a^^ 


IN  MEMORIAM 
FLORIAN  CAJORl 


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COLLEGE  ALGEBRA 


BY 

WILLIAM   BENJAMIN   FITE 

PROFESSOR   OF    MATHEMATICS    IX   COLUMBIA 
UNIVERSITY 


D.    C.    HEATH   &   CO.,  PUBLISHERS 
BOSTON  NEW  YORK  CHICAGO 


Copyright,  19 13, 
By  D.  C.  Heath  &  Co. 


PREFACE 

An  effort  has  been  made  to  present  here  the  elementary 
principles  of  algebra  in  a  simple  and  direct  way,  and  to  give 
rigorous  proofs  of  the  theorems  used.  It  seemed  desirable, 
however,  to  pass  over  certain  delicate  points  that  are  beyond 
the  comprehension  of  the  student,  and  to  omit  altogether  sev- 
eral difficult  proofs.  The  treatment  is  concrete,  and  graphical 
methods  have  been  used  freely. 

The  introduction  to  complex  numbers  that  is  commonly  given 
seems  to  me  to  be  arbitrary  and  unconvincing;  and  I  have 
sought  by  putting  forward  the  concrete,  geometrical  side  of  the 
question  and  by  careful  attention  to  the  proper  sequence  of 
the  ideas  introduced  to  present  these  numbers  in  a  rational  and 
convincing  way. 

Students  almost  invariably  think  that  the  symbol  oo  repre- 
sents a  number  called  "infinity,"  just  as  the  symbol  5  repre- 
sents a  number  called  "  five,"  and  this  view  is  fostered,  to  some 
extent  at  least,  by  the  conventional  use  of  the  word  "  infinity," 
and  by  the  employment  of  a  symbol  to  represent  it.  It  seemed 
worth  while,  therefore,  to  depart  from  this  conventional  treat- 
ment and  to  write  the  chapter  on  infinite  series  without  using 
either  the  symbol  for  infinity  or  the  word  itself,  except  as  it  is 
implied  in  the  phrase  "infinite  series." 

I  have  selected  the  problems  with  a  view  to  convincing  the 
student  that  algebra  is  a  body  of  principles  by  the  aid  of  which 
certain  kinds  of  important  information  can  be  obtained  from 
data  that  do  not  give  this  information  explicitly.  Many  prob- 
lems have  been  selected  that  illustrate  the  simpler  principles 
of  physics,  geometry,  and  analytic  geometry.  The  analytic 
geometry  problems  are  somewhat  of  an  innovation  in  a  text  of 

iii 


ivi306040 


iv  PREFACE 

this  kind,  and  objection  may  be  made  that  the  difficulties  of 
this  subject  ought  not  to  be  added  to  those  inherent  in  the 
algebra.  But  there  is  a  decided  advantage  in  making  the  stu- 
dent feel  that  mathematics  is  one. 

For  the  sake  of  adding  to  the  flexibility  of  the  book  two 
proofs  of  the  Binomial  Theorem  have  been  given,  one  in  the 
chapter  on  Permutations  and  Combinations,  and  the  other  in 
the  chapter  on  Mathematical  Induction.  Either  chapter  may 
be  omitted  without  the  necessity  of  omitting  the  Binomial 
Theorem. 

Attention  is  called  to  the  examples  given  in  the  text  of  the 
chapter  on  Mathematical  Induction  to  show  the  necessity  of 
both  parts  of  the  proof  by  induction. 

Articles  134-138  should  be  omitted  by  students  who  have 
not  studied  trigonometry. 

I  am  greatly  indebted  to  Professors  D.  R.  Curtis s  of  North- 
western University,  W.  B.  Carver  of  Cornell  University,  and 
R.  G.  D.  Richardson  and  H.  P.  Manning  of  Brown  University 
for  many  suggestions  of  importance  and  value. 


WILLIAM  BENJAMIN   FITE. 


Columbia  University, 
New  York  City. 


CONTENTS 

CHAPTEE  PA6K 

I.     The  Fundamental  Operations        ....  1 

11.     Factors  and  Multiples 19 

III.  Fractions .27 

IV.  Linear  Equations  in  One  Unknown       .        .        .38 
V.     Systems  of  Linear  Equations  in  Two  or  More 

Unknowns 47 

VI.     Fractional  and  Negative  Exponents.     Radicals  73  ^ 

VII.     Quadratics 84 

VIIL    Systems  of  Equations  in  Two  Unknowns  Solv- 
able BY  Means  of  Quadratics        .        .        .  110     "^ 

IX.    Progressions 123 

X.     Permutations  and  Combinations    ....  134 

XL    Mathematical  Induction 147 

XII.     Complex  Numbers 153*^^ 

XTII.     Theory  of  Equations 168 

XIV.     Determinants 201 

XV.     Inequalities 221 

XVI.    Partial  Fractions 228 

XVII.     Logarithms 238 

XVIIL     Variation 257 

XIX.     Infinite  Series 261 

Appendix 278 


COLLEGE  ALGEBRA 

CHAPTER   I 
THE   FUNDAMENTAL   OPERATIONS 

1.  The  operations  of  addition,  subtraction,  multiplication, 
and  division  are  called  the  fundamental  operations  of  algebra. 
We  shall  assume  that  the  student  is  familiar  with  the  details 
of  performing  these  operations  and  shall  confine  our  attention 
here  to  a  brief  consideration  of  their  important  properties. 

2.  The  representation  of  points  by  numbers.  —  Since  the 
letters  used  in  algebra  represent  numbers,  it  is  of  great  impor- 
tance for  the  student  to  have  a  clear  conception  of  the  nature 
of  numbers.  He  will  acquire  this  best  by  thinking  of  numbers 
as  representatives  of  points. 

Consider  an  indefinite  straight  line  called  the  axis,  a  fixed 

point  or  origin  0  on  this  line,  and  a  given  length ,  which 

we  shall  call  the  unit  length. 

? 4 1 

We  shall  let  the  point  A  which  is  on  the  line  at  a  distance 
to  the  right  of  the  origin  equal  to  this  unit  length  be  repre- 
sented by  the  number  1.  The  point  B  twice  this  distance  to 
the  right  of  the  origin  will  be  represented  by  the  number  2,  and 
similarly  any  point  to  the  right  of  the  origin  on  this  line  is 
represented  by  the  number  which  gives  its  distance  from  the 
origin  in  terms  of  the  given  unit  of  length.  The  points  to  the 
left  of  the  origin  on  the  line  are  also  represented  by  the  num- 
bers which  give  their  respective  distances  from  the  origin. 
But  it  is  necessary  to  distinguish  these  numbers  in  some  way 

1 


2  COLLEGE  ALGEBRA 

from  the  representatives  of  the  points  to  the  right  of  the  origin. 
This  is  done  by  prefixing  the  minus  sign  to  these  numbers,  and 
then  to  make  this  general  scheme  more  symmetrical,  we  should 
prefix  a  plus  sign  to  the  representatives  of  the  points  to  the 
right  of  the  origin.  Thus,  —  5  represents  the  point  five  units 
to  the  left  of  the  origin,  while  +  5  represents  the  point  five 
units  to  the  right  of  the  origin.  The  two  points  are  the  same 
distance  from  the  origin,  but  in  opposite  direction  from  it.  It 
is  this  difference  that  is  indicated  by  the  two  opposite  signs. 

Definitions.  —  The  numbers  that  represent  the  points  on  the 
left  of  the  origin  are  called  negative  numbers  and  those  that 
represent  the  points  on  the  right  of  the  origin  are  called  posi- 
tive numbers. 

The  plus  sign  that  is  a  part  of  the  positive  numbers  is  usually  not 
written  since  its  omission  causes  no  ambiguity. 

These  positive  and  negative  numbers,  together  with  the 
number  0,  which  represents  the  origin,  are  called  real  numbers 
for  a  reason  that  is  explained  in  Chapter  XII. 

The  quotient  of  two  integers  is  called  a  rational  number. 
Every  integer  is  a  rational  number  since  it  is  the  quotient  of 
itself  and  1. 

The  number  that  represents  a  point  is  called  the  abscissa  of 
this  point. 

The  distance  of  a  point  from  the  origin  is  called  the  absolute 
value  of  the  number  that  represents  the  point. 

Thus,  two  numbers  that  represent  points  on  opposite  sides  of  the  origin 
but  equidistant  from  it  have  the  same  absolute  value  ;  as,  for  example, 
the  numbers  +  7  and  —  7.     The  absolute  value  of  these  numbers  is  7. 

3.  Meaning  of  "greater  than"  and  "less  than."  —  If  the 
point  A  has  the  abscissa  a  and  is  to  the  right  of  the  point  B 
which  has  the  abscissa  b,  a  is  said  to  be  greater  than  b,  and  b 
less  than  a.  Thus  every  positive  number  is  greater  than  0  and 
every  negative  number  is  less  than  0.  Moreover  every  posi- 
tive number  is  greater  than  any  negative  number.     For  ex- 


THE  FUNDAMENTAL  OPERATIONS  3 

ample,  5  is  greater  than  —  6.  This  relationship  is  indicated 
thus ;  5  >  —  6,  or  —  6  <  5.  These  statements  are  read  "  5  is 
greater  than  —  6  "  and  "  —  6  is  less  than  5  "  respectively. 

We  express  the  fact  that  the  numbers  a  and  h  represent  dif- 
ferent points  by  the  symbol  a=^b,  which  is  read,  "  a  is  not 
equal  to  bJ' 

4.  Definition  of  sum.  —  If  ^  and  B  are  any  points  on  the 
axis  and  if  the  segment  AC  equals  the  segment  OB  in  magni- 
tude and  direction,  the  abscissa  c  of  the  point  O  is  called  the 
Bum  of  the  abscissae  a  and  b  of  the  points  A  and  B  respec- 
tively. 

f — ? ? — 4 


This  connection  of  the  numbers  a,  b,  and  c  is  expressed  thus : 
a  -f  6  =  c. 

The  process  of  finding  the  sum  of  two  numbers  is  called 
addition. 

By  the  sum  of  three  numbers  a,  b,  and  c  we  mean  the  sum 
of  a  -I-  &  and  c. 

If  we  represent  this  sum  by  a  +  6  +  c,  then  a  -}-  6  +  c  =  (a  +  6)  +  c. 

5.  Assumptions  concerning  real  numbers  and  the  operation 
of  addition.  —  We  shall  make  the  following  assumptions  con- 
cerning real  numbers  and  the  operation  of  addition.  A  careful 
inspection  of  a  figure  will  convince  the  student  of  the  appro- 
priateness of  most  of  these  assumptions. 

I.  If  a  and  b  are  any  numbers,  there  is  one  arid  only  one 
number  x  such  that  a-\-b  =  x. 

II.  Addition  is  commutative. 

This  means  that  a  -\-h  =  h  +  a. 

III.  The  sum  of  three  numbers  is  the  same,  iri'espective  of  the 
way  in  which  they  are  grouped. 

Thus,  a  +  6  +  c  =  (a  +  6)  -f-  c  =  a  +  (?>  +  c). 

This  is  the  associative  law  of  addition. 


4  COLLEGE  ALGEBRA 

IV.  Ifa-\-b  =  a-\-c,  then  b  =  c. 

This  is  the  law  of  cancellation  for  addition. 

V.  There  is  one  and  only  one  number'  x  such  that  a;  +  ic  =  a?. 
This  number  is  0. 

VI.  For  every  number  a,  there  is  one  and  only  one  number 
(—a)  such  that  a  +  (—  a)  =  0. 

If  a  is  a  negative  number,  it  is  clear  that  the  (—  «)  of  this  assumption 
must  be  a  positive  number. 

The  familiar  statement  that  if  equal  numbers  be  added  to 
equal  numbers  the  sums  are  equal  numbers  is  equivalent  to 
Assumption  I. 

6.  Consequences  of  the  assumptions.  —  The  following  theo- 
rems are  consequences  of  these  assumptions. 

1.  If  a  is  any  number,    a  +  0  =  a. 

2.  If  a  and  b  are  any  number s,  there  is  one  and  only  one  num- 
ber X  such  that  a=b-\-x. 

Definition.  —  The  process  of  finding  x  when  a  and  b  are  given 
is  called  the  process  of  subtracting  b  from  a.  In  this  process 
a  is  called  the  minuend,  b  the  subtrahend,  and  x  the  difference 
or  remainder. 

Subtraction  is  indicated  by  the  minus  sign. 

Thus,  the  relation  of  «,  h,  and  x  just  described  is  expressed  by  the 

equation 

a  —  h  =  X. 

The  process  of  subtracting  b  from  a  is  equivalent  to  that  of 
finding  the  length  and  direction  of  the  segment  BAj  A  and  B 
being  the  points  whose  abscissae  are  a  and  b  respectively. 

The  statement  that  if  equal  numbers  be  subtracted  from 
equal  numbers  the  remainders  are  equal  numbers  is  equivalent 
to  Theorem  2. 


THE  FUNDAMENTAL  OPERATIONS  5 

3.  The  result  of  adding  (  —  b)  to  any  number  a  is  the  same  as 
the  result  of  subtracting  b  from  a. 

That  is,  a+(-&)=a-6. 

4.  The  result  of  subtracting  (—6)  from  any  number  a  is  equal 
to  the  result  of  adding  b  to  a. 

That  is,  a  -  (—  6)  =  a  +  h. 

Although  Theorems  1-4  are  consequences  of  Assumptions 
I-V,  they  seem  so  obvious  from  the  definition  of  addition  and 
an  inspection  of  a  figure  that  we  shall  omit  the  formal  proofs. 

The  familiar  rule  for  finding  the  difference  of  two  numbers, 
—  namely,  to  change  the  sign  of  the  subtrahend  and  add  the 
resulting  number  to  the  minuend,  —  is  an  immediate  conse- 
quence of  Theorems  3  and  4. 

7.  Parentheses.  —  An  expression  like  a  —  b-\-c-\-d  indi- 
cates that  (—6)  is  to  be  added  to  a,  c  added  to  the  resulting 
sum,  and  d  to  this  last  sum.  That  is,  a  — 6  + c  +  cZ  is  a  sim- 
plified form  for  \_{a  —  b)  -\-  c'] -{-  d.     Now 

[(a  _  6)  +  c]  +  d  =  (a  -  6)  +  (c  +  d)  (III) 

=  a-|-[-6  +  (c  +  <i)] 
=  a-{-\_—b-\-c  +  d']. 
Hence,     x-{-{a  —  b-{-c-\-d)  =  x-\-(a-\-[—b-'rc-{-  dj) 

=  (x  +  a)  +  {-b  +  c  +  d) 
=  (x  +  a  —  b)^-{c-\-d) 
=  {x-\-a  —  b-{-c)-\-d 
=  x-\-a—b-{-c-\-d. 

This  procedure  is  obviously  applicable  to  parentheses  con- 
taining any  number  of  terms.     Hence, 

5.  An  expression  involving  parentheses  preceded  by  the  plus 
sign  is  unchanged  if  the  parentheses  are  removed  and  the  inclosed 
terms  are  left  with  their  original  signs. 


6 


COLLEGE  ALGEBRA 


Consider  the  expression 

a—  (b-\-c  —  d). 
If  a—  (b-\-c  —  d)  =  x, 

then  a  =  x-{-(b-{-c  —  d),  (Definition  of  Subtraction.) 

and  a  =  x-\-b-{-c  —  d.  (5) 

If  we  subtract  b  and  c  from  each  member  of  this  equation 
and  add  d  to  each  member,  we  get 

a  —  b—c-\-d  =  x. 
Hence,  a—  (b-\-c  —  d)  =  a  — b  —  c-{-d. 

This  procedure  is  obviously  applicable  to  any  parentheses 
preceded  by  the  minus  sign.     Hence, 

6.  An  expression  involving  parentheses  preceded  by  the  minus 
sign  is  unchanged  if  the  parentheses  are  removed  and  the  sign  of 
each  of  the  inclosed  terms  is  changed. 

8.  Definition  of  multiplication.  —  We  have  seen  that  the  real 
numbers  are  the  abscissae  of  the  points  on  an  indefinite  line 
with  reference  to  a  given  origin  and  a  given  unit  distance. 

Let  this  line  be  X'X,  the 
origin  O,  and  the  unit  dis- 
tance  OU. 

Let  Y'  Y  be  an  indefinite  line 
through  0  perpendicular  to 
X'X.  The  real  numbers  will 
also  be  the  abscissse  of  the 
points  of  this  line  with  refer- 
ence to  the  same  origin  and  the 
same  unit  distance,  the  positive 
numbers  representing  the  points 
above  the  origin. 

Let  a  and  b  be  any  two  numbers,  and  let  A  be  the  point  of  X^X  whose 
abscissa  is  a,  and  B  the  point  of  T'y  whose  abscissa  is  b.  Draw  UB  and 
then  AC  parallel  to  UB.  The  number  c  which  is  the  abscissa  of  the  point 
C  on  Y'Y  is  called  the  product  of  a  and  6. 

This  connection  of  a,  6,  and  c  is  expressed  thus  : 

a-b  =  c;  or  thus,  a  x  b  =  c;  or  thus,  ab  =  c. 


THE  FUNDAMENTAL  OPERATIONS  7 

The  process  of  finding  the  product  of  two  numbers  is  called 
multiplication,  and  the  numbers  are  called  factors. 

By  the  product  of  three  numbers  a,  b,  and  c  we  mean  the 
product  of  ab  and  c. 

If  we  represent  this  product  by  abc,  then  abc  =  (ah)c. 

9.  Assumptions  concerning  real  nimibers  and  the  operation 
of  multiplication.  —  We  shall  make  the  following  assumptions 
concerning  real  numbers  and  the  operation  of  multiplication. 
A  careful  inspection  of  a  figure  will  convince  the  student  of 
the  appropriateness  of  most  of  these  assumptions. 

VII.  If  a  and  b  are  any  numbers,  there  is  one  and  only  one 
number  x  such  that  ab  =  x. 

VIII.  Multiplication  is  commutative. 
This  means  that  ab  =  ba. 

IX.  The  product  of  three  numbers  is  the  same  irrespective  of 
the  way  in  which  they  are  grouped. 

Thus,  (ab)c  =  a(bc). 

This  is  the  associative  law  of  multiplication. 

X.  Ifab  =  ac,  and  a  ^  0,  then  b  =  c. 

This  is  the  law  of  cancellation   for  multiplication. 

XI.  The  number  1  is  such  that  1  •  1  =  1. 

XII.  For  every  number  a  not  equal  to  0  there  is  one  and  only 
one  number  a'  such  that  a  •  a'  =  1. 

The  number  a'  is  called  the  reciprocal  of  a,  and  a,  the  recipro- 
cal of  a'. 

XIII.  Multiplication  is  distributive  as  to  addition. 
This  means  that  a(b  +  c)  =  ab  -\-  ac. 

XIV.  The  product  of  two  positive  numbers  is  a  positive 
number. 


8 


COLLEGE  ALGEBRA 


The  familiar  statement  that  if  equal  numbers  be  multiplied 
by  equal  numbers  the  products  are  equal  numbers  is  equivalent 
to  Assumption  VII. 

10.  Consequences  of  the  assumptions.  —  The  following 
theorems  are  consequences  of  these  assumptions : 


l.Ifa  is  any  number,     a  •  1 
This  is  obvious  from  the  figure. 


a. 


S.Ifa  and  b  are  any  numbers  and  b  is  not  equal  to  0,  there  is 
one  and  only  one  number  x  such  that  a  =  bx. 

Let  TJ  and  B  be  the  points  of  X' X 
whose  abscissae  are  1  and  h  respectively, 
and  let  A  be  the  point  of  Y'Y  whose 
abscissa  is  a.  Then  if  we  draw  the  line 
through  U  parallel  to  BA,  it  will  cut  Y'  Y 
in  a  point  C  whose  abscissa  x  is  such  that 
a  =  bx.  Moreover,  there  is  no  other  num- 
ber y  such  that  a  =  by.  This  can  also 
be  seen  as  follows : 

If  there  were  another  number  y  such 

that  by  =  a,  we  should  have  bx  =  by.    But 

it  follows  from  the  law  of  cancellation  for  multiplication  that  this  relation 

can  be  true  only  when  x=  y.     Hence  the  number  a;  is  the  only  number 

that  satisfies  the  condition  of  the  theorem. 


Definition  of  division.  —  The  process  of  finding  x  when  a  and 
b  are  given  is  called  the  process  of  dividing  a  by  b.  In  this 
process  a  is  called  the  dividend,  b  the  divisor,  and  x  the 
quotient. 

Division  is  indicated  by  writing  the  dividend  over  the  divi- 
sor with  a  horizontal  line  between  them,  and  also  by  the 
symbol  -^ . 

Thus,  -  indicates  the  division  of  a  by  6,  as  does  also  the  symbol  a  -j-  ft. 
b 

The  relation  a  =  6a;  is  then  equivalent  to  the  relation  -  = «,  provided 

b 
that  b  is  not  0. 


THE  FUNDAMENTAL  OPERATIONS  9 

9.  Multiplication  is  distributive  as  to  subtraction. 
For  if  a  —  b  =  x, 

then  a  =  x+b,     (Definition  of  Subtraction.) 

and  ma  =  mx  +  mb.  QLUT) 

Hence,  ma  —  7nb  =  7nx  =  m(a  —  b). 

(Definition  of  Subtraction  and  VII.) 

10.  The  product  of  0  and  any  number  is  0,  and  conversely,  if 
the  product  of  two  numbers  is  0,  one  of  the  numbers  must  be  0. 

This  can  be  seen  directly  from  the  figure. 

In  the  definition  of  division  it  was  stipulated  that  the  divisor 
should  not  be  0.  The  reason  for  this  is  clear  from  Theorem  10. 
Since  ic  •  0  =  0  for  all  values  of  x,  there  can  be  no  value  of  x  for 

which  X  •  0  =  a,  where  a  is  different  from  0.    Hence  ^  does  not 

0 

represent  any  number  when  a  is  net  0.     On  the  other  hand,  - 

might,  with  our  definition  of  division,  consistently  stand  for 
any  number. 

The  student  should  therefore  be  careful  never  to  attempt  to 
divide  by  0  because  there  is  no  such  operation  as  division  by  0. 

It  follows  from  Theorem  10  that  -  =  0,  for  all  values  of  b  dif- 

b 
ferent  from  0. 

11.  Tlie  prodtict  of  a  positive  number  by  a  negative  number  is 
a  negative  number. 

Let  a  and  b  be  any  positive  numbers. 
Then  there  is  a  number  x  such  that 

a{-b)=x,  (VII) 

and  therefore      a{—b)-\-ab  =  x-{-ab.  (I) 

a  [(-  6)  +  6]  =  a;  +  ab.  (XIII) 

a-0  =  x  +  ab.  (VI) 

0  =  a;  +  a6.  (10) 

0  —  ab  =  x.     (Definition  of  Subtraction.) 


10  COLLEGE  ALGEBRA 

0-\-(-ab)  =  x.  (3) 

(-ab)  =  x.  (1) 

Hence,  a{—b)  =  {—  ah). 

But  ah  is  a  positive  number  (XIV)  and  therefore  (—ah)  is 
negative. 

12.    Tlie  product  of  two  negative  numbers  is  a  positive  number. 
Let  (—a)  and  (—  5)  be  any  negative  numbers. 
Then  there  is  a  number  x  such  that 

(-a)(-6)=a.,  (VII) 

and  therefore     (—a) (—6)  -^a{—b)  =  x-\-a{—b). 
(  —  6)  [  ( —  a)  4-  a]  =  X  —  a6. 
(-b)  '0  =  x-ab. 
0  —  x  —  ab. 
0  -\-ah  =  X. 

ah  =  x.  (1) 

Therefore  {—a){—b)  =  ab. 

Moreover  ah  is  a  positive  number.  (XIV) 

The  well-known  rule  of  signs  for  multiplication  follows  from 
XIV,  11,  and  12. 

.  13.  The  result  of  dividing  a  positive  number  by  a  positive 
number,  or  a  negative  number  by  a  negative  number,  is  a  positive 
number. 

14.  Tlie  result  of  dividing  a  positive  number  by  a  negative 
number,  or  a  negative  number  by  a  positive  number,  is  a  negative 
number. 

The  proofs  of  13  and  14  depend  upon  XIV,  11,  12,  and  the  definition 
of  division.     The  details  are  left  as  exercises  for  the  student. 

11.  The  removal  and  insertion  of  parentheses. —  The  rule  for 
removing  single  parentheses  is  contained  in  Theorems  5  and  6. 
Two  or  more  parentheses,  no  one  of  which  is  contained  within 
another  one,  can  be  removed  simultaneously  by  the  same  rule. 
When  parentheses  are  contained  within  parentheses,  it  is  best 


THE  FUNDAMENTAL  OPERATIONS  U 

to  proceed  step  by  step,  at  each  step  removing  the  innermost 
parentheses.  In  such  cases  more  than  one  pair  of  parentheses 
can  be  removed  at  a  time  if  care  is  exercised  to  get  the  signs 
correct. 

If  the  expression  in  parentheses  is  to  be  multiplied  by  a 
monomial  factor,  every  term  in  the  parentheses  should  be  mul- 
tiplied by  this  factor.     This  follows  from  Assumption  XIII. 

The  rule  for  the  insertion  of  parentheses  is  an  immediate  consequence 
of  the  rule  for  their  removal.  We  leave  the  formulation  of  it  as  an  exer- 
cise for  the  student. 

EXERCISES 

Simplify  the  following  expressions : 

1.  2-[3  +  5-7J3-(5  +  2)-lj4-l]. 

2.  l4-2|3-5  +  2(-64-2)J. 

3.  4:a-3-(x~3a)-\-(2a-x). 

4.  2x^y-3(x' -hf)-\-2(xy^-xh/). 

5.  6a;-2[3a;  +  4-3(a:-l)]. 

6.  8m-[-(w- n)  +  (3  7?. -6m)]. 

7.  6?>-5(/.-2)-f6[6-(5  +  3Z>)+7]. 


8.  4a-(-3a-[-4a-J-2a-3a-f  i;-10a]). 

9.  7  X  -{-  2[3  X  -  \2  X  -(y  -  2  x)l  -  2(x  -  8  y)]. 


10.  a-(-a-S-7a-3[a-3a-4]H-3J-[3a-a  +  l]). 

11.  a(x  +  y)-  h{3x-2y). 

12.  2a(p?  -  W) -  h  [a52  _  3  (4  a^  -  5  W)']. 

13.  x-lx-lx-ix-l)]]. 


14.  3  m  -  2  ?i[l  -f-  3  m  (1  -  m  4-  w)]. 

15.  2{x  +  y  +  z)-3(x-y-\-z)+2{x^-y-z). 

16.  ^ah-3h\a-\-h-2{3a-2b)\. 

17.  a(a;  4-  ?/)  -f  h{x  -  y)-  x{a  -f  6)  -f  y{a  -  b). 


12  COLLEGE  ALGEBRA 

18.    (a  +  6  — c)  +  (a  — 6  +  c)  +  (— a  + 6  4-c). 


19.   4m  +  2  Jn  — 3(m  —  n)J— 3m  —  271. 


20.  Sm  —  2n  —  2\n  —  3{m  —  n)\  —  4:m. 

In  the  following  polynomials  inclose  all  the  terms  containing 
x^  in  parentheses  preceded  by  the  plus  sign  and  inclose  the 
terms  containing  x  in  parentheses  preceded  by  the  minus  sign : 

21.  4:  x^ -{- ax^  —  3  X -\- bx  +  5. 

22.  x^  +  2ax  +  mx^  —  2x^3-\-kx. 

23.  hx  +  x'^ -{■  ex  +  \. 

24.  2  a'W  -2h'^x-  aV  +  x^. 

25.  —  ax  -\-  a?  —  hx  —  2  X  —  ^. 

26.  7  ic  —  6  ic'^  +  4  a  —  3  6x  —  c:i(?. 

27.  ax^  +  hx  +  c-3ix?  -lOx  +  4.-  hxK 

28.  3  +  a  +  ic  +  £c'^  —  mx  —  na;^  —  vic. 

29.  —  3  a;  +  5  —  aa^^  +  fta;'^  4-  cx^  —  ax  —  bx  —  ex. 

30.  3(ax^  —  2bx-^c)—5(a-\-cx  —  bx'^). 

12.  Addition  and  subtraction  of  polynomials.  —  The  problem 
of  expressing  the  sum  of  the  polynomials  3x— 4:y-\-6z  —  2, 
—  5x  —  6y-[-z,  and  3x-\-2y-{-z-\-5m  the  simplest  possible 
form  is  the  same  as  the  problem  of  simplifying  the  expression 

(3x-4.y-\-6z-2)  +  {-5x-6y-\-z)-^(3x-\-2y-\-z-\-5). 

Now  this  is  equal  to 

Sx  —  4:y  +  6z-2  —  5x-6y-{-z-{-3x-\-2y  +  z-h5 
=  3x—5x  +  3x-4:y—6y-{-2y-^6z-\-z-\-z-2-\-5 
=  x—Sy  +  Sz-}-3. 

The  usual  rule  for  finding  the  sum  of  these  polynomials  is 
merely  a  more  direct  way  to  this  same  result.  And  in  general 
the  rule  for  finding  the  sum  of  any  polynomials  is  a  direct 


THE  FUNDAMENTAL  OPERATIONS  13 

consequence  of  Theorem  5  and  Assumption  II.  The  rule  for 
simplifying  the  sum  of  two  or  more  similar  terms  is  a  conse- 
quence of  Assumption  XIII. 

We  get  the  rule  for  subtracting  one  polynomial  from  another  in  a  simi- 
lar way  by  using  Theorem  6  instead  of  Theorem  5. 

EXERCISES 

1.  Add  7  a3  _  4  a  +  2  a2  -  5,  -2  a  +  5  a^ -\-l  -  2  a%  and 
4  -I-  3  a  +  2  a^  +  a'. 

2.  Add  —  5oiy^y  —  2  xy^  —  o?  —  y^^  a^  -\-  y^,  a^  —  y^,  and 
a^  +  Sa^y-j-Sxy^  +  y\ 

3.  Subtract  S  a^ -\' S  a'b  -  5  ab'' +  2b^  from  a' -  6  a^ft  + 
4  aft^  -f-  3  b^,  and  then  add  the  subtrahend  to  the  result.  Can 
you  predict  what  the  final  result  will  be  ? 

4.  Subtract  x^  —  y^  from  a^  -{-  y^. 

5.  Subtract  aj^  -f  2/^  from  a^  —  y^. 

6.  From  the  sum  of  y"^  —  S  y  -\-  2,  7  y  -  2  +  y"^,  and  6  2/^  -f 
6y  -\-  6  subtract  y^  -\-  5  y  +  6. 

7.  From  the  sum  of  4:  a^b  A- 5  ab%  5  a^ -2  ab^ -hSb%  and 
2a'-2b^-{-Sa'b-i-7ab^  subtract  7  a^ -{- 7  a^b -{- 10  ab^ -^  b  . 

8.  Add  {r-^s-\-t)x+ (r  —  s  +  t)y-\-(r-{-s  —  t)z,  (2r  —  s)x 
•i- {s -\- 3t)y -\- {r -{■  s -{-t)Zj  and  {s-\-3t)x-{-(2r-\-4:t)y-h{s-{-3t)z. 

9.  From  the  sum  of  ma^ -{•  3  na^ -\- px  —  q  and  nx^  —  mx^-^ 
2qx+p  subtract  (m  -\-n)a^-\-  {m  —  3n)x^-{-{p-\-q)x-\-  (q—p). 

10.  Subtract  (a  +  b)x-\- {b-\-c)y +  (G  +  a)z  iiom  (c-\-a)x-\- 
{a-{-b)y -^{b-{-c)z. 

11.  Subtract  {c -{- a)x -\- {a -\-b)y -{-  (b -{- c)z  from  (a-{-b)x-\- 
{b  +  c)y-{-{c  +  a)z. 

12.  Does  a  consideration  of  the  results  in  Exs.  10  and  11 
suggest  to  you  any  general  principle  ?  Consider  also  Exs.  4 
and  5. 


14  COLLEGE  ALGEBRA 

13.  Add  a~\-b  —  c,  a  —  b-\-c,  and  —a-\-b-\-c.  (See  Ex.  18, 
§11). 

14.  Subtract  2(a-\-b  —  c)  from  a  -f  6  -}-  c. 

15.  From  the  sum  of  a^  +  3  a-b  +  3ab-  +  b^  and  a^  —  S  a^b  + 
3  ab^  —  b^  subtract  the  sum  of  a^  -f  b^  and  a^  —  b'^. 

16.  Subtract  a^  +  b^  from  a'^  +  3 a^ft  +  3 a^^  +  ?)3^  and  a^-b^ 
from  a3-3a26  +  3a&2-63, 

17.  Is  there  any  difference  between  the  result  in  Ex.  15  and 
the  sura  of  the  results  in  Ex.  16  ?  Could  you  have  answered 
this  question  without  going  through  the  details  of  Exs.  15 
and  16  ? 

18.  Subtract  ax^-\-bx  +  c  from  dx~  -\-ex-\-f. 

19.  Is  the  result  of  subtracting  a  number  from  zero  the 
same  as  that  of  subtracting  zero  from  this  number  ? 

20.  What  must  be  added  to  7  a^  —  8  a^  +  4  a  4-  7  in  order 
that  the  sum  shall  be  2  a^  +  4  a^  +  6  a  - 1  ? 

21.  What  must  be  added  to  A  afy  —  6  xy^  +  8  /  in  order  that 
the  sum  shall  be  5  x^ -{- 7  xy^ -\- y^  ? 

13.  Multiplication  of  polynomials.  —  The  rule  for  forming 
the  product  of  two  polynomials  is  a  direct  consequence  of 
Assumption  XIII. 

Consider  the  product  of  the  two  binomials  a  +  6  and  x-^y. 
By  Assumption  XIII 

(a  -{-b)(x  +  y)  =  (a+6)  x  +  (a+b)y=ax-\-bx+ay+by. 

The  product  of  any  two  polynomials  can  be  treated  step  by 
step  in  a  similar  way. 

Homogeneous  polynomials.  —  A  polynomial  all  of  whose 
terms  are  of  the  same  degree  is  said  to  be  homogeneous. 

Thus,  5  ic2  -f  7  ?/2  and  a^  -f-  3  a^b  +  3  a&2  -f  68  are  homogeneous  polyno- 
mials of  degrees  2  and  3  respectively. 


THE  FUNDAMENTAL  OPERATIONS 


15 


The  product  of  two  homogeneous  polynomials  is  a  homo- 
geneous polynomial  whose  degree  is  equal  to  the  sum  of  the 
degrees  of  the  factors.  This  fact  is  useful  in  checking  the 
work  of  forming  the  product  of  two  homogeneous  polynomials. 


EXERCISES 

Simplify  the  following  indicated  products  and  use  the  fore- 
going check  whenever  it  is  applicable : 

1.  {x  +  2y){x  +  Sy). 

2.  (a +  5)  (a -8). 

3.  {x^  +  2xy  +  f){x'-2xy  +  f). 

4.  (x^-2xy  +  y''){x'-^x^y-{-^xy^-f). 

5.  {p?  +  ab-\-h-){a''-ah  +  h-). 

6.  (a  +  ar  +  ai-^  +  ai-^  -f-  ar^)  (1  —  r). 

7.  (a  -  h)  {a*  +  a^b  +  a%^  +  aW  +  h"). 
.8.  {x^y  +  zjix  +  y-z). 

9.  (a  +  6  -f  c)  (2  a  -  6  -f  3  c). 

•     10.  (a2  +  6a  +  4)(a3-4a2  +  5a-f 2). 

11.  {a  +  h^x  —  y){a  —  h  —  x-\-y). 

12.  (3  a^  -  5  a^ft  +  7  a&2  -I-  4  6^  (4  a^  4- 10  a6  +  5  62) . 

13.  (x^  -\-  y"^  -^  z^  ^-2  xy  -\-2  yz  -\-2  zx){x  +  y  -^  z). 

14.  (2a-36  +  5)(2a-36-4). 

15.  (wi*  +  m~n^  -f-  n^)  (m*  —  m^n^  +  n*) . 

16.  (c3  +  9(r-f-27c  +  27)((r-6c  +  9). 

17.  (16-Sx  +  4:i^-23(^  +  x')(2-\-x). 

18.  (r'  +  5r~4)(j'2  +  5r-3). 

19.  (7ft2-(-56^-9c2)(6a2-46'-^  +  c2). 

20.  (aj2  +  3a;-f 9)(x2-3a;-f 9). 


16  COLLEGE  ALGEBRA 

14.  Special  products.  —  The  following  special  products 
should  be  memorized  by  the  student  in  order  that  he  may  be 
able  to  write  down  products  of  these  types  from  an  inspection 
of  the  factors.  He  should  verify  each  of  the  formulae  by 
actual  multiplication.  He  should  also  translate  each  one  into 
ordinary  language. 

1.  {a  +  hy  =  a'-{-2ab-\-h\ 

2.  (a^hf  =  a^  +  3  a?h  +  Z  ay  +  h\ 

3.  {a-\-h){a-h)  =  a''-h\ 

4.  (a  +  &  +  c)2  =  a^  +  62  4-  c^  +  2  a&  +  2  6c  +  2  ca. 

EXERCISES 
Write  the  results  of  the  following  indicated  multiplications : 

1.  {a-Vf,  11.  {x^2f. 

2.  {a-hf.  12.  (5  a -4)1 

3.  {2x  +  3yf.  13.  (10 +  9  6)  (10 -9  6). 

4.  (2x  +  ^yf.  14.  (a  +  iy. 

5.  (2a;  +  3?/)(2a;-32/).  15.  (J  x  +  10y)(10  y -7  x). 

6.  (3  a -6)2.  16.  {^m  +  n  +  r)\ 

7.  (a-6-c)2.  17.  (a^  +  a^  +  l). 

8.  (2  a; -3  2/ +  4  2)2.  18.  {^r-sf. 

9.  (a  +  6  +  c)(a  +  6-c).  19.  {s-^rf. 

10.    (2  m  — 5  w)  (2  m  — 5  w).  20.    {x-\-y —  z){x  —  y -^z). 

15.  Division  of  poljrnomials,  —  If  ^  and  B  are  two  polyno- 
mials such  that  the  degree  of  A  in  some  letter  is  equal  to,  or 
greater  than,  the  degree  of  B  in  this  letter,  the  process  of  find- 
ing two  polynomials  Q  and  i?  such  that 

A  =  Bq^R, 

and  R  is  of  lower  degree  in  the  given  letter  than  B,  is  called 
the  process  of  dividing  A  by  B.  In  this  process  A  is  called 
the   dividend,  B  the  divisor,  Q  the  quotient,  and  B  the  re- 


THE  FUNDAMENTAL  OPERATIONS  17 

mainder.  If  R=0  we  say  that  A  is  divisible  by  B,  or  that 
the  division  is  exact.  If  the  divisor  is  a  monomial,  it  follows 
immediately  from  Assumption  XIII  that  the  quotient  is  the 
sum  of  the  partial  quotients  obtained  by  dividing  each  term 
of  the  dividend  by  the  divisor.  How  the  quotient  is  obtained 
when  the  divisor  contains  more  than  one  term  is  best  shown 
by  an  example. 

Divide  12  x^  —  7  x^y  —  14  xy"^  +  5  y^  by  4  x  —  5  y. 

12  ic3  -    1  x^y-  14  xy^  +  5  y^lix  -by 

12  x»  -  15  x'^y 1 3x^4- 2x2/ -2/2 

8  x'-^y  —  14  xy'^ 
8  x^y  -  10  xy^ 

—  4  xy2  +  5  2/3 

—  4  xy"^  +  5  1/3 

We  know  that  the  required  quotient  is  Sx"^  +  2xy  —  y^  and  that  the 
remainder  is  0,  since  the  products  of  the  successive  terms  of  the  quotient 
and  the  divisor  when  subtracted  in  turn  from  the  dividend  give  a  final 
remainder  of  0. 

Before  beginning  this  work  of  finding  the  successive  terms  of  the  quo- 
tient, the  student  should  see  to  it  that  the  dividend  and  the  divisor  are 
arranged  according  to  the  descending  powers  of  some  letter,  although 
when  the  division  is  exact  they  may  be  arranged  according  to  the  ascend- 
ing powers. 

The  relation  A  =  BQ-\-B, 

holds  true  for  all  values  of  the  letters  involved  and  is  for  this 
reason  called  an  identity.     (See  §  31.) 

EXERCISES 
Perform  the  following  indicated  divisions : 

1.  (a^-b'-)^(a-b), 

2.  (x'  -j-aff  ^i/)  -ir  (x'  -{-xy  ^y"), 

3.  Q^-f)^(x-y). 

5.    (a*-|-4a3  +  6a2  +  4a-hl)-^(a2-f 2a-f  1). 


18  COLLEGE  ALGEBRA 

6.  (a^4-3a^  +  7a2  +  8ci  +  6)-v-(a2  +  a-f-3). 

8.  (r' -  s') --- {r -\- s).  9.    (a^-h')  ^  (a +  b). 

10.  [(a  _  by-  5(a  -  6)  +  6]  -  [(a  -  ?>)  -  3] . 

11.  (m^  4-7^3) -J- (m- 71).  12.    (a;2+ 5  aj  + 6)-- (.^•  +  2). 
13.  (o5_?>5)h-(«-^6).                   14.    {a'-h')-r-  (a-\-b). 

15.  (64  +  9?>2  +  81)-(62  +  36+9). 

16.  (x''  +  5x  +  4)^(x''  +  Sx  +  2). 

17.  (a^  +  9  a^fe  +  27  ab^+  27  ?>3)  -j-  (a2  4.  e  a6  -f  9  62). 

18.  (a^  +  9  a26  +  27  a^^  _^  27  6^)  _^(«  4.  3  5). 

19.  (x^-y^)-^(x-y).  20,    (aj4_^4)^(^^,^)^ 


CHAPTER   II 

FACTORS   AND    MULTIPLES 

16.  The  problem  of  factoring  a  polynomial  —  that  is,  of  find- 
ing two  polynomials  whose  product  is  the  given  polynomial  — 
is,  in  general,  more  difficult  than  that  of  dividing  one  polyno- 
mial by  another  one,  since  in  factoring  both  the  divisor  and 
the  quotient  are  unknown.  There  are,  however,  certain  types 
of  polynomials  that  can  be  factored  readily.  The  more  com- 
mon of  these  and  their  factors  are  given  here. 

17.  1.  Polynomials  with  monomial  factors.  —  It  follows  from 
the  distributive  law  for  multiplication  that  a  factor  of  all  the 
terms  of  a  polynomial  is  a  factor  of  the  polynomial. 

For  example,  Ix^y'^  is  a  factor  of  every  term  of  the  polynomial 
6  x^ip-  +  8  x'^y^  —  10  x^y3,  and  is  therefore  a  factor  of  the  polynomial. 

6  ic3y2  +  8  x4«/3  -  10  5C52/8  =  2  xhf^(^  +  4  a-y  -  5  xhf). 

2.  Polynomials  that  can  be  factored  by  grouping  their  terms. 

—  The  following  is  a  typical  polynomial  of  this  form : 

ax -\-  ay  -{■  az  ■\- hx  ■\- hy  -\-  hz  —{ax  +  ay  +  az)  +  (&x  +  6y  +  hz) 
=  «(»  +  ?/  +  2;)  +  &(a-  +  2/  +  iS!) 
=  (x  +  ?/  f  ^)(a  +  &). 

3.  The  difference  of  two  squares. 

a2-62^  (a+6)(a-6). 

Many  polynomials  that  are  more  complicated  in  appearance  than  this 
one  are  in  reality  of  this  form.     For  example, 

a2  -f  2  a6  +  62  _  a;2  _|.  2  a:?/  -  y2 

=  (a2  +  2  ah  +  l)^)-ix?- -2xy  +  y'^) 
=  (a-\-hy-(x-yy 
=  (a-{-b  +  x  —  y){a-\-b  —  x  +  y). 
19 


20  COLLEGE  ALGEBRA 

4.  The  sum  of  two  cubes. 

a^  +  b^  =  (a-i-  6)(a2-  ab-^i^). 

5.  The  difference  of  two  cubes. 

6.  Trinomial  squares. 

a'^  +  2ab-hb^=(a-h  by. 
a2  -  2  a&  +  ft2  =  (a  -  by. 

7.  The  quadratic  trinomial  ax"^  -i-bx+  c. 

If  four  numbers  k,  I,  m,  and  n  can  be  found  such  that  km  =  a, 
Im  +  kn  =  b,  and  Z?i  =  c,  then  aa;^  +  5^  +  0  =(A:x  +  l)(mx  +  n). 

8.  Polynomials  of   four  terms  that  are  the    cubes   of  bi- 
nomials. 

a^-\-Sa''b-[-Sat^+b^  =  (a+  by. 

a3  _  3  flSft  +  3  a62  _  iy3  =(a  _  5)3. 

9.  The  sum  of  two  like  odd  powers. 

If  n  is  an  odd  positive  interger  a**  +  6"  is  divisible  by  a  +  6. 

10.   The  difference  of  two  like  powers. 

For  every  positive  integral  value  of  n,  a"  —  6**  is  divisible 
by  a  —  6. 

This  statement  and  the  one  under  9  can  be  proved  by  means  of  the 
factor  theorem  (§  142,  Cor.) 

18.  Whether  a  given  polynomial  can  be  factored  or  not 
depends  somewhat  upon  our  point  of  view.  Thus,  the  polyno- 
mial a^  —  2  has  the  factors  x  +  V2  and  x  —  V2  although  it  has 
none  with  rational  numerical  coefficients.  Also  x^  —  y  has  the 
factors  X  +  'Vy  and  x  —  ^y,  but  none  that  are  free  from 
radicals. 

In  this  chapter  we  shall  confine  our  attention  to  polynomials  whose 
numerical  coeflBcients  are  integers,  and  when  we  speak  of  the  factors  of 
a  polynomial  we  shall  mean  factors  that  are  free  from  radicals  and  that 
have  integral  numerical  coefficients. 


FACTORS  AND  MULTIPLES  21 

Definition.  —  A  polynomial  that  has  no  factors  of  this  kind 
is  said  to  be  prime. 

A  polynomial  whose  numerical  coefficients  are  integers  with 
a  common  factor  greater  than  1  is  not  prime  since  this  numeri- 
cal factor  is  a  factor  of  the  kind  just  described. 

The  work  of  factoring  can  be  checked  by  making  use  of  the 
fact  that  the  product  of  all  the  factors  found  should  equal  the 
given  polynomial. 

EXERCISES 

Factor  the  following  polynomials  into  their  prime  factors : 

1.  12  a'b'-\- IS  a'b'- 21  a'b'.  16.  a2-5a  +  4. 

2.  (a +  6)2-1.  17.  m^  — 25  m2  —  m  + 5. 

3.  Sx^-\-12x^-j-6x-{-l.  18.  -2x^-\-Sxy  +  27y\ 

4.  9a2  +  24a6c+16  6V.  19.  x* -\- 34:  xY -^  225  y*. 

5.  27-64a^.  20.  a^ +4.a^ +  2  a-\-S. 

6.  ^9-x^-\-4:xy-4:f.  21.  54:  a^x^  -  16  b^x^. 

7.  a2-f4a-77.  22.  6x'^-x-15. 

8.  a^  —  b^  —  a  +  b.  23.  a^  —  x*. 

9.  a^  — 2a3  +  l.  24.  7n^ -\- 2  xy  —  x'^  —  y\ 

10.  x^ -\- x"^ -\- X -\- 1.  25.  x^  —  y\ 

11.  a'b  —  aV.  26.  x^  +  y^. 

12.  2xy-x^~y\  27.  15  a"  ^- 4:1  ab  +  2S  b\ 

13.  Q?-3x'^y  +  3x7f-y^-z\  28.  {x-{-yf-21. 

14.  12o}-{-13ab-35b\  29.  Qx^ +  3  x" -\-2  x-^1, 

15.  6a^2_,_7^2/-20  2/2.  30.  a^  +  6  a^  +  12  a  + 8. 

19.  Highest  common  factor.  —  A  polynomial  that  is  a  factor 
of  each  of  two  or  more  polynomials  is  called  a  common  factor 
of  these  polynomials.  A  given  set  of  polynomials  may  have 
several  common  factors  of  the  same  degree. 

Thus,  2  a;2  _  4  0^  4-  2  2^2  and  4  x^  —  4  y^  have  the  common  factors  x  —  y^ 
—  x  +  y,  2x  —  2y,  and  —  2  x  +  2  y  of  the  first  degree. 


22  COLLEGE  ALGEBRA 

Among  all  the  common  factors  of  the  highest  degree  of  two 
or  more  polynomials  there  is  always  one  the  greatest  common 
divisor  of  whose  numerical  coefficients  is  equal  to  the  greatest 
common  divisor  of  all  the  numerical  coefficients  of  the  given 
polynomials.  We  call  this  a  highest  common  factor  of  the 
polynomials. 

Thus,  2x  —  2y  isa,  highest  common  factor  of  the  polynomials  given  in 
the  preceding  illustration. 

But  —  2  ic  +  2  ?/  is  also  a  highest  common  factor  of  these 
polynomials ;  and  in  general,  any  set  of  polynomials  has  two 
highest  common  factors  which  differ  only  in  sign.  In  most 
cases  it  makes  no  difference  which  of  these  two  highest  com- 
mon factors  of  the  given  polynomials  we  take,  and  it  is  there- 
fore customary  to  speak  of  "  the  highest  common  factor,"  as  if 
there  were  but  one. 

If  the  given  polynomials  can  be  factored  into  their  prime 
factors,  their  H.  C.  F.  (highest  common  factor)  can  be  deter- 
mined immediately  by  inspecting  these  prime  factors. 

20.  Lowest  common  multiple.  —  A  polynomial  that  is  divisi- 
ble by  each  of  two  or  more  polynomials  is  called  a  common 
multiple  of  these  polynomials.  A  given  set  of  polynomials 
has  many  common  multiples  of  the  same  degree. 

Thus,  2  x^  —  4:  xy  -\-  2  y"^  and  4  x^  —  4  y2  have  the  common  multiples 
4  x^—  4  x^y  —  4  xy^  +  4  2/^,  8  x^  —  8  x^y  —  8  xy'^  +  8  y^,  and  many  others 
which  are  of  the  third  degree. 

Among  all  the  common  multiples  of  the  lowest  degree  of  two 
or  more  polynomials  there  is  always  one  the  greatest  common 
divisor  of  whose  numerical  coefficients  is  equal  to  the  least 
common  multiple  of  the  greatest  common  divisors  of  the 
numerical  coefficients  of  the  respective  polynomials.  We  call 
this  a  lowest  common  multiple  of  the  polynomials. 

Thus,  4x^-4x^y-4xy^  +  4 
two  polynomials  given  in  the  preceding  illustration. 


FACTORS  AND  MULTIPLES  23 

Every  set  of  polynomials  has  two  lowest  common  multiples 
which  differ  only  in  sign.  The  expression  "the  lowest  com- 
mon multiple  "  refers  to  either  one. 

If  the  given  polynomials  can  be  factored,  their  L.  C.  M. 
(lowest  common  multiple)  can  readily  be  formed. 

EXERCISES 

Find  the  H.  C.  F.  and  the  L.  C.  M.  of  the  following  poly- 
nomials : 

1.  25,30,40. 

2.  '3?  —]^^x^—  y^. 

3.  a2  +  2a6-hZ>2,  a^  +  3  a6 -j- 2  61 

4.  h'  +  f,  W-f. 

5.  x^  +  2/2,  y^—  i/,     x^  —  y\ 

6.  a^-243,  a;2-6a;  +  9,  a;2-9. 

7.  ax  —  hx  —  ay  -\-  by,  a^-\-ab  —  2  b\ 

8.  a2  4-3  a- 28,  Sa^-  20  a. 

9.  a^  —  x^,  a^  —  a^,  a^  —  x"^. 

10.  x''-\-x-6,  2x'^-\-2x  —12. 

11.  a''-¥,  ¥-a\ 

12.  x^  +  l  x  +  12,  aj2  4_  5  X  -I-  6. 

13.  6  a2  -  13  a5  +  6  b\  10  a^  _  13  aft  -  3  b\ 

14.  x^  +  2/^  ^^  ~~  y^- 

15.  24  m2  -14.  mn-  20  n^,  24  m^  +  6  mw  -  45  ?^2. 

16.  16a4-81ft^   8a3-27  6^ 

17.  aj2  _|_  6  icy  -}-  9  ?/^  a^  +  9  a;2^  +  27  a^?/^  +  27  f. 

18.  ao^  +  2  aa;2  -j-  3  aa;  +  6  a,  4  aa;  +  8  a. 

19.  15  a2  + 11  a6  +  2  62,  5  a^  -f-  7  a6  +  2  b\ 

20.  a-\-b,  a-\-2b,  a  +  3  6. 


24  COLLEGE  ALGEBRA 

21.  Euclid's  Method  for  finding  the  highest  common  factor 
of  two  or  more  polynomials.  —  If  we  cannot  find  the  factors  of 
the  given  polynomials,  the  preceding  method  cannot  be  used. 
In  such  a  case,  if  the  polynomials  contain  only  one  letter,  we 
can  fall  back  on  the  following  method,  which  is  called  Euclid's 
Method. 

Let  A  and  B  be  any  two  polynomials  in  x  whose  H.  C.  F.  we 
wish  to  find.  Arrange  them  according  to  the  descending  powers 
of  X,  and  if  the  degree  of  B  is  not  greater  than  the  degree  of  A 
divide  A  by  B. 

Denote  the  quotient  by  Qi  and  the  remainder  by  Bi.  Then 
(§15) 

A  =  BQ,-{-B„ 

or  A-BQi  =  R^. 

Now  any  factor  of  J5  is  a  factor  of  BQi.  Hence  any  common 
factor  of  A  and  J5  is  a  factor  of  ^  —  BQi,  which  is  the  same 
as  Ri ;  and  is  therefore  a  common  factor  of  B  and  i?i.  Con- 
versely, any  common  factor  of  B  and  J?i  is  a  factor  ot  BQi-\-  Bi, 
which  is  the  same  as  A ;  and  is  therefore  a  common  factor  of 
A  and  B.  The  highest  common  factor  of  A  and  B  must  accord- 
ingly be  the  same  as  the  highest  common  factor  of  B  and  J?i. 
We  can  then  shift  the  problem  to  the  finding  of  the  highest 
common  factor  of  B  and  B^.  Now  R^  is  of  lower  degree  in  x 
than  B.  We  therefore  proceed  as  before  and  divide  B  by  R^. 
If  the  quotient  is  Q2  and  the  remainder  R2,  we  have 

B=R,Q2  +  R2, 
or  B-RiQ2  =  R2. 

Reasoning  similar  to  that  used  before  shows  that  the  highest 
common  factor  of  B  and  Ri  is  the  highest  common  factor  of  Ri 
and  i?2- 

By  continuing  in  this  way  we  can  establish  the  following 
series  of  identities : 


FACTORS  AND  MULTIPLES  25 

El  =  i^aQs  +  Bsf 


Bk  —  B,,^iQk+2  +  B^^2i 

in  which  Bk+2  tloes  not  contain  x.  This  is  possible  since  each 
B  is  of  lower  degree  in  x  than  the  preceding  one,  and  if  the 
process  is  continued  far  enough  we  must  come  to  an  B  that 
does  not  contain  x. 

It  is  clear  from  the  method  of  formation  of  these  equations 
that  the  highest  common  factor  of  B^  and  B^+i  is  also  the  high- 
est common  factor  of  A  and  B.  If  now  Bk+2  =  0,  B,,+i  is  the 
highest  common  factor  of  Bj,  and  i?t+i  and  therefore  also  of  A 
and  B.  But  if  Bk+2  ^  0,  ^  and  B  have  no  common  factor  con- 
taining x. 

Example.  —  Find  the  H.  C.  F.  of  x^  +  2  a;2  _  13  ^  +  10  and  x^  + 
x:^  -  10  X  +  8. 

These  two  polynomials  are  of  the  same  degree  and  either  may  there- 
fore be  taken  as  the  divisor.     We  select  the  second  one. 

x^-^2x^-13x  +  10  \  a^  +    x2  -  10  a;  -H  8 
x»  +     x^-10x-\-    S     1 

x2-    Sx  +  ~Y\  x^+    a;2-10x  +  8 
x  +  4  x3-3x2+2x 


4x2-12x  +  8 
4a;2_i2a;+8 


Here  i?2  =  0  and  therefore  i?i  =  a;2  -  3  x  +  2  is  the  H.  C.  F. 

In  applying  this  process  the  given  polynomials  should  first  be  divided 
by  any  monomial  factors  they  may  have  and  account  taken  of  these 
factors  in  the  final  result.  Fractional  coefficients  should  be  avoided  by 
introducing  or  removing  numerical  factors  w^henever  necessary  at  any 
stage  of  the  process.  This  can  always  be  done  in  such  a  way  as  not  to 
affect  the  final  result. 


26  COLLEGE  ALGEBRA 

22.  Lowest  common  multiple  of  two  polynomials.  —  If  F  is 
the  highest  common  factor  of  the  polynomials  A  and  B,  and 
A  =  A'F,  B  —  B'Fj  then  A'  and  J3'  have  no  common  factor. 

Now =  AB^  —  A'B.     Hence is  a  common  multiple  of 

A  and  B.  Suppose  now  that  M  is  any  common  multiple  of  A 
and  B.  Then  we  should  have  M=  AQ  =  A'FQ.  But  M  must 
also  be  divisible  by  B,  which  is  the  same  as  B'F.  Hence  Q 
must  be  divisible  by  B'.  In  order  that  M  be  the  lowest  com- 
mon multiple,  Q  must  be  the  same  as  B'.     Hence  the  L.  C.  M. 

AB 
of  A  and  B  is  A'FB'  =  AB'  = .     This  is  equivalent  to  say- 
ing that 

The  lowest  common  multiple  oftivo  polynomials  is  equal  to  their 
product  divided  by  their  highest  common  factor. 

EXERCISES 

Find  the  H.  C.  F.  and  the  L.  C.  M.  of  the  following  poly- 
nomials : 

1.  a^  -  a;2  +  3  a;  -f  5,  and  y^+^x^-^x-h. 

2.  a;^-f-4a^  +  5aj2  +  4aj  +  l  and  a;^  +  5  a;^  -f  3  x^  -  10  a;  -  4. 

3.  6  a:^  _  22  a;2  +  15  a;  -  2  and  6  a;4  -  a-3  +  4  a;2  -  13  aj  +  6. 

4.  2  a:^  _j_  12  a;2  +  19  a;  +  3  and  2  a^  -f  13  aj2  _^  17  a!  +  3. 

5.  a;^  -f  3  a^  4-  6  a;2  +  3  a?  -  5  and  a;4  +  a^  -  11  a;2  -  10  a;  + 10. 

6.  a^  -f-  6  a;2  +  11  a;  -f  6  and  a.-3  +  8  a)2  +  11  a;  -  20. 

7.  2  a^  +  4  a3  -  2a2  -  4  a  and  a^  4-  2  a2  -  3. 

8.  ?/4  +  3  2/3  +  4  2/2  -  3  2/  -  5  and  ?/^  +  3  ?/3  4-  6  /  +  3  2/  +  5. 

9.  a'  +  3  a2  -f-  8  a  -f  6  and  a^  4-  3  a^  -f  9  a2  +  8  a  +  6. 

10.  62  _  e  5  4.  2  and  W  -'lb''  -22h  +  8. 

11.  a;5  _|.  4  ^  ^  3.2  _  5  ^  _^  1  and  a;^  4-  a;^  ^  5  ^j^  +  5  a;  4-  5. 


CHAPTER   III 

FRACTIONS 

23.  Definition  and  principles.  —  A  fraction  is  the  indicated 
quotient  of  two  numbers. 

Thus,  2     -3     _L4      V2^ 

3'     5    '     -.5        3 

The  principles  upon  which  the  usual  operations  with  fractions 
rest  can  be  readily  derived  from  the  assumptions  of  Chapter  I. 
These  principles  are : 

1.  Tlie  value  of  a  fraction  is  not  changed  by  multiplying  the 
numerator  and  the  denominator  by  the  same  number. 

Consider  the  fraction  - .     It  follows  from  the  definition  of 
b 
division  that  cl 

a  =  b  •  -. 
b 

Hence,  ak  =  bk  --. 

'  b 

But  this  is  only  another  way  of  saying  that 
ak_a 
bk~b' 

This  holds  for  all  values  of  k  except  0,  and  since  division  by  a  number 
is  equivalent  to  multiplication  by  the  reciprocal  of  this  number,  we  have 
also  proved  that  the  value  of  a  fraction  is  not  changed  by  dividing  the  nu- 
merator and  the  denominator  by  the  same  number. 

2.  Tlie  sign  of  a  fraction  is  changed  if  the  sign  of  either  the 
numerator  or  the  denominator  is  changed. 

This  is  an  immediate  consequence  of  the  law  of  signs  for  division, 
which  is  derived  from  the  assumptions  and  the  definition  of  division. 

3.  The  sum  of  two  fractions  loith  a  common  denominator  is 
a  fraction  whose  numerator  is  the  sum  of  the  numerators  of  the 
given  fractions  and  whose  denominator  is  the  common  denominator. 

27 


28  COLLEGE  ALGEBRA 

Let  the  given  fractions  be  -  and  - . 

c  c 

^ow  a  =  C'  -, 

c 

and  b  =  c  '    .  (Definition  of  division.) 

c 

Hence,      a  +  ?>  =  c-  4-  c- 
c        c 

=  c  (  -  +  - ).      (Distributive  law  for  multiplication.) 
\c      cj 

From  this  it  follows  that 

^^±^  =  -  +  - .  (Definition  of  division.) 

c         c      c 

In  a  similar  way  we  see  that 

a—h_a      b 

c  c      c 

In  view  of  what  has  just  been  proved  we  can  say  that  divi- 
sion is  distributive  as  to  addition. 

This  includes  the  statement  that  division  is  distributive  as  to  subtrac- 
tion since  subtraction  is  included  under  addition.     (See  3,  Chapter  I.) 

a         '  c 
4.    The  sum  and  the  difference  of  any  two  fractions,  -  a7id  -, 

0  a 


uiK  eyaut/  vu   — 

bd 

bd      "^"^"^ 

For  by  1, 

a  _ad 
b^bd' 

and 

c_bc 
d~bd' 

Hence  by  3, 

a 
b' 

c     ad-\-bc 

d  ~      bd     '■ 

and 

a 
b 

c     ad  — be 
d        bd 

FRACTIONS  29 

5.  The  product  of  two  fractions  is  a  fraction  whose  numerator 
is  the  product  of  their  numerators  and  whose  denominator  is  the 
product  of  their  denominators. 

-ci  j^    a     c  c 

For,  ^' ^'-^  =  ^'-J^ 

h    d  d 

(Definition  of  division,  and  IX,  Chapter  I.) 

and  hd  '  -^  .  ~  =b  •  -  •  -  •  d=  a*  -  >  d  =  ac. 

b     d  b     d  d 

(Definition  of  division,  VIII,  and  IX,  Chapter  I.) 

Hence,  -  •  -  =  — .  (Definition  of  division.) 

'  b     d     bd  ^  ^ 

6.  Tlie  quotient  resulting  from  dividing  one  fraction  by  another 
is  a  fraction  luhose  numerator  is  the  product  of  the  numerator  of 
the  dividend  and  the  denominator  of  the  divisor  and  whose  de- 
nominator is  the  product  of  the  denominator  of  the  dividend  and 
the  numerator  of  the  divisor. 

If  the  dividend  is  -  and  the  divisor  - ,  we  have,   from  the 
b  d 

definition  of  division,  ^ 

a_b    c 

b^c'  d 

d 

a 

Hence,  ad  =  -'  be, 

d 

a 
ad^h 

be      c* 
d 


and 


When  we  are  dealing  with  more  than  two  fractions  the  principles  to  be 
employed  are  obvious  extensions  of  those  just  developed. 


30  COLLEGE  ALGEBRA 

The  following  reductions  are  applicable  to  rational  fractions  ; 
that  is,  fractions  which  are  equivalent  to  indicated  quotients  of 
polynomials. 

24.  Reduction  of  certain  fractions  to  mixed  expressions. — 

If  the  degree  of  the  numerator  of  a  fraction  in  a  certain  letter 
is  equal  to,  or  greater  than,  the  degree  of  the  denominator  in 
this  letter,  we  can  arrange  both  numerator  and  denominator 
according  to  the  descending  powers  of  this  letter  and  divide 
the  numerator  by  the  denominator.  Then,  if  the  given  frac- 
tion is  -,  we  have  the  identity 

a  ,-r 

6  =  ^  +  6' 

where  g  is  a  quotient,  and  r  the  remainder,  of  this  division. 
Here  g  is  a  polynomial  and  -  is  a  fraction  whose  numerator  is 

of  lower  degree  than  its  denominator.  Hence,  if  the  degree  of 
the  numerator  of  a  fraction  in  any  letter  is  equal  to,  or  greater 
than,  the  degree  of  the  denominator  in  this  letter,  the  fraction  is 
equal  to  the  sum  of  a  polynomial  and  a  fraction  whose  7iumerator 
is  of  lower  degree  in  this  letter  than  its  denominator. 

25.  Reduction  of  a  fraction  to  its  lowest  terms.  —  A  fraction 
is  said  to  be  in  its  lowest  terms  when  its  numerator  and  de- 
nominator have  no  common  factor. 

A  fraction  can  be  reduced  to  its  lowest  terms  by  dividing 
the  numerator  and  denominator  by  their  highest  common 
factor  (§  23,  1). 

26.  Reduction  of  several  fractions  to  equivalent  fractions 
with  a  common  denominator  of  lowest  possible  degree.  —  Find 
the  L.  C.  M.  of  the  denominators,  which  is  called  the  lowest 
common  denominator  of  the  fractions,  and  divide  this  by  the 
denominator  of  each  fraction.  Then  multiply  the  numerator 
and  the  denominator  of  each  fraction  by  the  corresponding 
quotient.     The  resulting  fractions  are  equivalent  to  the  orig- 


FRACTIONS  31 

inal  ones  (§  23,  1)  and  have  the  required  common  denominator. 
It  is  unnecessary  actually  to  perform  the  multiplication  of 
each  denominator  by  the  corresponding  quotient,  since  we 
know  beforehand  that  the  product  will  be  the  lowest  common 
denominator. 

27.  Addition  and  subtraction  of  fractions.  —  In  order  to  add 
or  subtract  fractions,  first  reduce  them  to  equivalent  fractions 
with  a  common  denominator  and  then  proceed  in  accordance 
with  §  23,  3. 

In  order  to  add  a  polynomial  to  a  fraction,  consider  the 
polynomial  as  a  fraction  with  the  denominator  1  and  then  pro- 
ceed as  directed. 

28.  Multiplication  and  division  of  fractions.  —  The  rules  for 
multiplying  and  dividing  fractions  are  given  in  §  23,  5,  6. 
The  given  fractions  should  be  reduced  to  their  lowest  terms 
before  applying  these  rules. 

EXERCISES 
Reduce  the  following  fractions  to  mixed  expressions : 


a  +  b 
a^-1 


ar^  +  2a^- 

a; +  4 

a^-a^-^2 

X-4: 

x'-hl 

x'-l 

4:X 

ar  — 1 

3.    -J^.  6.           ^ 

a  — 1  x-^  1 

Reduce  the  following  fractions  to  their  lowest  terms: 

^    18  xy  ^^    m^  +  5m^4-2m  +  10 

12x2/"*  *         m2-f7m  +  10 

8      ^^-^  11  ^-(y-h^y 

x''-6x-{-9  '   (x  +  yf-z^ 

^    a;5_y5  ^^    a^  +  4aj2-f-8a;  +  5 


y^-a^  a^-Sx-'+Sx-^-T 


32  COLLEGE  ALGEBRA 

Perform  the  following  indicated  operations  and  bring  youi 
results  to  their  simplest  forms  : 

13.    ^-  +  ^_.  15.  ^  ^'^ 


K  +  l     a-l  0^-20  +  1     a^  —  1 

14.   -i ^  +  -?-  16.         '"  +  ^      .+       ^  +  3 


K  +  l     »  — 1     s  +  S  a;^+5x  +  e     ii?+l  x+12 


(x -!)(«- 2)      {x-2){x-Z)      (»-3)(x-l) 

18.  m^-m  +  1— !^.  21.   2aW.10a^ 

m  + 1  5  3?y*       3  6^ 

19.  *+-^-  o„   n^yw 


25. 


a      bj\a      bj\        a  —  hj\        a  —  b 

w}  +  mn  +  mr  +  ****  .  w?  —  t^ 
m"^  —  mn  —  mr  +  nr     m^  —  n^ 

10i»  +  25 


-  (>+Mh^ 


3x2 


ym^  +  71^      m'^  —  n^ 
28.1  i+l^        1 


\  ^  /m  -\-  n  .m  —  n\ 
J     \m  —  n      m  +  nj 


1      r^jT^-r} 


30.  ^~^ .         ^^y        . 

3  a;       a;2  -|-  3  a;^/  —  4  2/2 


FRACTIONS  33 


Simplify  the  following : 

31.    ^i^.  34.       "^ 


m  N_ 

^  R  35. 


H 


1  + 


1  + 


5508 

29.  Ratio. — The  fraction  -  is  called  the  ratio  of  a  to  h.    The 

h 

numerator  a  is  called  the  antecedent  of  the  ratio  and  the  de- 
nominator h  the  consequent. 

Two  numbers  a  and  6  are  said  to  be  commensurable  if  there 

is  a  number  r  such  that  -  and  -  are  integers.    If  we  call  these 

integers  x  and  y  respectively,  that  is,  if  -  =  a;  and  -  =  y,  then 

r  r 

-  =  -•  Hence  the  ratio  of  the  two  commensurable  numbers  is 
h     y 

a  rational  fraction.     Not  all  numbers  are  commensurable. 

For  example,  V2  and  1  are  incommensurable,  since  if  they  were  com- 
mensurable we  should  have  ^^  =  -  or  2  =  —  •     Hence,  2  y2  =  ^2.    But 

1         y  y-^ 

ever}''  prime  factor  of  x^  and  of  y^  occurs  to  an  even  power  and  this  rela- 
tion requires  that  2  occur  as  a  factor  of  x^  to  a  power  one  higher  than 
the  power  to  which  it  occurs  in  y^. 

30.  Proportion.  —  The  statement  that  two  fractions  are  equal 
is  called  a  proportion.  The  numerators  and  denominators  of 
the  fractions  are  called  the  terms  of  the  proportion.     Thus,  «, 

b,  c,  and  d  are  the  terms  of  the  proportion  -  =  -•     This  propor- 

b      d 
tion  is  sometimes  read  "a  is  to  6  as  c  is  to  d"      The  first 
and  the  last  terms  are   called  the  extremes,  and  the  other 
two  the  means,  of  the  proportion. 


34  COLLEGE  ALGEBRA 

If  we  clear  the  equation  -  =  - 
b      d 

of  fractions,  we  get  ad  —  he. 

Hence,  in  any  proportion  the  product  of  the  means  is  equal  to 
the  product  of  the  extremes. 

The  student  can  verify  that  if 
a_c 
b~d' 

then  ^  =  i 

c     d 

and  -  =  - . 

a     G 

The  former  of  these  two  is  said  to  have  been  obtained  from 
the  original  one  by  alternation  and  the  latter  by  inversion. 

Suppose  that  -==-  =  -  =  r. 

^^  h     d     f 

Then  a  =  hr, 

c  =dr, 

and  e  =fr. 

Hence,  a  +  c  -^  e  =  br  -\-  dr  -\-  fr  =  r  (b  -{-  d  -\-  f), 

-I  a  +  c  +  e  ace 

and  ,       ,      ^=  r  = "  =  -  =  -,. 

b  +  d^f  b     d     f 

If  we  had  started  with  more  than  three  equal  ratios,  it  is 
easy  to  see  that  an  analogous  procedure  would  have  led  to  an 
analogous  result.  We  conclude  therefore  that  in  a  series  of 
equal  ratios  the  sum  of  the  antecedents  is  to  the  sum  of  the  conse- 
quents as  any  antecedent  is  to  its  consequent. 

If  -  =  -,/  is  called  the  fourth  proportional  to  a  and  b  and  c. 

If  —  =— ,  m  is  called  a  mean  proportional  between  a  and  b. 
m      b 

If  -  =  -,  ^  is  called  the  third  proportional  to  a  and  b. 
b     t 


FRACTIONS  35 

EXERCISES  AND  PROBLEMS 

1.  In  the  diatonic  music  scale  the  notes  O,  D,  E,  F,  G,  A, 
and  B  are  produced  respectively  by  c,  d,  e,  /,  g,  a,  and  b  vibra- 
tions of  the  air  per  second,  where  -  =  -^  =•<--  =  -  and  -  =  —  = 

e     b     a     5  g     2  d 

—  =  -  •  Find  the  number  of  vibrations  per  second  that  pro- 
2c      6  f  ^ 

duce  each  of  these  notes,  it  being  assumed  that  C  is  produced 

by  256  vibrations  per  second. 

2.  If  the  terms  of  a  ratio  are  positive  and  the  same  positive 
number  is  added  to  each,  in  what  way  is  the  value  of  the 
ratio  changed? 

Let.  the  ratio  be  -  and  let  x  be  the  positive  number.     Then  com- 
pare the  values  of  -  and  ^-±-?. 
b  b  +  X 

Theorem  from  Geometry In  a  right  triangle  the  perpendicular 

dropped  from  the  vertex  of  the  right  angle  to  the  hypotenuse 
is  a  mean  proportional  between  the  segments  into  which  it 
divides  the  hypotenuse. 

3.  The  hypotenuse  of  a  right  triangle  is  20  inches  long  and 
the  perpendicular  from  the  vertex  of  the  right  angle  upon 
the  hypotenuse  is  8  inches  long.  Where  is  the  foot  of  this 
perpendicular  ? 

4.  Prove  that  if  -=  -,  then 

b      d 

(a) 


(&) 
(c) 


a-\-b 
b 

c-\-d 
d 

a-b 
b 

c-d 
d 

a-{-b 

c-\-d 

a—b      c—d 


Proportion  (a)  is  said  to  have  been  derived  from  the  given  proportion 
by  composition,  (6)  by  division,  and  (c)  by  composition  and  division. 


36 


COLLEGE  ALGEBRA 


5.  Find  the  mean  proportionals  between  16  and  36. 

6.  Find  the  third  proportional  to  15  and  28. 

7.  Find  the  fourth  proportional  to  13,  28,  and  36. 

Theorem  from  Geometry.  —  A  perpendicular  dropped  from  any 
point  in  the  circumference  of  a  circle  to  a  diameter  is  a  mean 
proportional  between  the  segments  into  which  it  divides  the 
diameter. 

8.  A  perpendicular  drawn  from  a  point  in  the  circumference 
of  a  circle  of  radius  6  inches  to  a  diameter  divides  this  diameter 
into  two  parts,  the  ratio  of  whose  lengths  is  equal  to  ^.  Find 
the  length  of  this  perpendicular. 

9.   In  the  triangle  ABC,  AB  =  S,  BC  =  17, 

and  CA  =  20.  Through  what  point  of  BC 
must  a  line  DE  parallel  to  AB  be  drawn  in 
order  that  DE  =  ^AB? 

DE^EC 
AB     BC' 


Hint,      ±^  = 


A  property  of  similar  triangles.  —  The    areas   of 
similar    triangles    are    proportional    to    the 


squares  of  two  corresponding  sides. 

Thus,  if  S\  and  82  are  the 
areas  respectively  of  the  similar 
triangles  ABC  and  DEF,  then 

Si^AB^  ^BC^  ^CJ^. 
S2     DE^     EF^     Flf 


10.   If    AB  =  9    inches, 


what  must   be   the  length 

of  DE  in  order  that  the  triangle  DEF  be  twice  as  large  as  the 

triangle  ^B(7? 

11.   If   Si  =  20  square   inches,  xS'2  =  25   square   inches,  and 
JBC=4  inches,  how  long  is  EF? 


FRACTIONS 


37 


12.  In  the  triangle  ABC  the  line  DE  is  to  be 
drawn  parallel  to  AB  in  such  a  way  that  the  tri- 
angle cut  off  shall  be  one  third  of  ABC.  At 
what  point  in  ^4 (7  must  Z)  be  ? 

Formula  from  Physics.  —  If   two   falling   bodies 

pass  over  the  distances  Sj  feet  and  Sg  feet  in  ti  seconds  and  ^2 

s      t^ 
seconds  respectively,  then  —  =  -^. 

,  ^2     ^2 

13.  What  is  the  ratio  of  the  distances  passed  over  by  the 
two  bodies  if  the  second  one  falls  three  times  as  long  as  the 
first  one  ? 

14.  If  a  body  falls  64  feet  in  2  seconds,  how  far  will  a  body 
fall  in  5  seconds  ? 

Theorem  from  Geometry.  —  If  a  plane  be 
drawn  parallel  to  the  base  of  a  pyramid 
V-ABC,  cutting  the  pyramid  in  the  sec- 
tion DEF,  and  if  we  denote  the  areas  of 
ABC  and  DEF  by  JSi  and  S2  respec- 
tively, then 

where  VH  and  VK  are  the  distances  of 
the  base  and  the  cutting  plane  respec- 
tively from  the  vertex. 

15.  In  a  pyramid  whose  altitude  is  10  inches,  how  far  from 
the  vertex  must  a  cutting  plane  be  in  order  that  the  section 
DEF  be  three  fourths  as  large  as  the  base  ? 

16.  If  the  area  of  the  base  of  the  pyramid  is  14  square 
inches,  and  the  altitude  is  9  inches,  what  is  the  area  of  the 
section  midway  between  the  base  and  the  vertex  ? 


CHAPTER   IV 

LINEAR  EQUATIONS   IN   ONE   UNKNOWN 

31.  Definition  of  equation.  —  The  statement  that  two  alge- 
braic expressions  are  equal  is  called  an  equation. 

If  the  statement  contains  only  one  letter  whose  value  is 
unknown,  it  may  be  considered  as  a  description  of  a  number. 
A  number  answers  this  description  if  the  two  expressions  have 
the  same  value  when  this  number  is  substituted  for  the  letter 
whose  value  is  unknown. 

An  identity  is  a  description  that  fits  every  number. 

Thus,  the  two  members  of  the  identity,  (cc  —  1)2  =  a:'2—  2  x  +  1,  are 
equal  to  one  another  regardless  of  what  value  is  substituted  for  x. 

An  identity  is  sometimes  indicated  by  the  symbol  =  . 

Thus,  (x  -  1)2  =  a;2  -  2  X  +  1. 

On  the  other  hand,  the  two  members  of  the  equation 
^x  -\-l  =  X  —  1  do  not  have  the  same  value  for  most  values 
of  X. 

When  x  =  2,  for  example,  the  left  member  equals  7  and  the  right 
member  1.  As  a  matter  of  fact,  —  1  is  the  only  value  of  x  that  makes 
the  two  members  of  this  equation  equal ;  that  is,  —1  is  the  only  number 
that  answers  the  description  given  by  this  equation. 

An  equation  that  does  not  fit  every  number  is  called  an 
equation  of  condition,  because  it  imposes  a  condition  on  the 
numbers  that  satisfy  it. 

If  each  of  its  members  is  a  polynomial  in  the  letter  repre- 
senting the  number  described,  the  equation  is  called  a  rational 
integral  equation.  If  neither  of  these  polynomials  is  of  degree 
greater  than  1  in  the  letter  representing  the  number  described, 
the  equation  is  said  to  be  of  the  first  degree,  or  linear. 

38 


LINEAR  EQUATIONS  IN  ONE   UNKNOWN  39 

A  number  that  answers  the  description  given  by  the  equation 
is  said  to  satisfy  the  equation,  and  is  called  a  root  of  the  equa- 
tion. The  process  of  finding  the  number  or  numbers  described 
by  an  equation  is  called  the  process  of  solving  the  equation. 

At  the  opposite  extreme  from  identities  stand  descriptions 
(equations)  that  do  not  fit  any  number. 

For  example,  2x  +  l  =  5a;  +  4  —  3a;, 

32.  Equivalent  equations.  —  Two  equations  that  describe  the 
same  numbers  are  said  to  be  equivalent. 

The  equations  (a;  —  1)  (a;  -  2)=  0  and  {x  -  \Y{x  -  2)  =  0  each  de- 
scribe the  numbers  1  and  2  ;  but  they  are  not  considered  equivalent  since 
the  latter  in  a  sense  describes  the  number  1  twice.     (See  §  148.) 

The  following  principles  concerning  equivalent  equations 
are  of  frequent  application  in  the  solution  of  equations. 

1.  If  we  add  the  same  number  to  each  member  of  an  equation 
and  equate  the  sums,  the  equation  thus  formed  is  equivalent  to  the 
original  one. 

If  we  represent  the  left  member  of  the  equation  by  L  and 
the  right  member  by  R,  we  have 

L  =  R.  (1) 

Either  L,  or  R,  or  both  of  them  contain  x. 
Let  the  new  equation  be 

L  +  8  =  R  +  S.  (2) 

Now  any  number  described  by  equation  (1),  when  put  in 
place  of  cc,  makes  L=R,  and,  therefore,  L-\-S=R-\-S  (I,  Chap- 
ter I).  Moreover,  any  number  described  by  equation  (2),  when 
put  in  place  of  £c,  makes  L-\-S  —  R-\-  8,  and,  therefore,  L==  R 
(IV,  Chapter  I).  That  is,  the  two  equations  describe  the  same 
number  and  are  therefore  equivalent. 

The  case  of  the  subtraction  of  the  same  number  from  each  member  of 
the  equation  is  included  in  this  proof  since  the  subtraction  of  any  number 
/S'is  equivalent  to  the  addition  of  (—  S)  (3,  Chapter  1). 


40  COLLEGE  ALGEBRA 

2.  If  we  multiply  or  divide  both  members  of  an  equation  by  the 
same  number  and  equate  the  results,  the  equatio7i  thus  formed  is 
equivalent  to  the  original  one,  provided  that  the  number  by  which 
we  multiply  or  divide  is  not  zero  and  is  not  expressed  in  the  terms 
of  the  uyiknown  number  of  the  equation. 

Let  the  original  equation  be 

L  =  R,  (1) 

and  the  new  one  LM=  RM.  (2) 

Now  any  number  described  by  equation  (1)  when  put  in  place 
of  ic,  makes  L  =  Ry  and  therefore  LM—  RM  (VII,  Chapter  I). 

Moreover,  any  number  described  by  equation  (2),  when  put  in 
place  of  X,  makes  LM=RM,  and  therefore  L=R  (X,  Chapter  I). 

That  is,  the  two  equations  describe  the  same  numbers  and 
are  therefore  equivalent. 

The  argument  just  given  covers  the  case  of  dividing  both  members  by 
the  same  number,  since  to  divide  a  number  by  M  is  equivalent  to  multi- 
plying it  by  J-. 
M 

If  the  M  of  the  demonstration  were  0,  equation  (2)  would  be 

0  =  0. 

But  this  is  an  identity  and  is  not  equivalent  to  (1). 

The  reason  for  saying  that  the  number  by  which  we  multiply 
or  divide  the  two  members  of  the  equation  shall  not  be  ex- 
pressed in  terms  of  the  unknown  of  the  equation  will  be  clear 
from  the  following  examples : 

Consider  the  equation       2,x  +  S  =  x  +  2. 

If  we  multiply  each  member  by  x  —  3  and  equate  the  products,  we  get 

(2  5c  +  3)(a;  -  3)  =  (x  +  2)(x  -  3). 
Now  the  student  can  readily  verify  that  -  1  is  the  only  number  that 
answers  the  first  description,  whereas  both  —  1  and  3  answer  the  second 
one.     Hence  the  two  equations  are  not  equivalent  —  the  second  one  has  a 
root  which  is  not  a  root  of  the  first  one. 
Consider  now  the  equation 

x^-9  =  x  +  S. 


LINEAR  EQUATIONS  IN  ONE  UNKNOWN  41 

If  we  divide  both  members  by  a-  4-  3  and  equate  the  quotients,  we  get 

The  student  can  verify  that  4  and  —  3  answer  the  first  description  and 
that  4  is  tlie  only  number  answering  the  second  one.  Hence  the  two 
equations  are  not  equivalent. 

33.  From  Assumption  I  it  is  clear  that  we  can  transpose 
any  term  from  one  member  of  the  equation  to  the  other  pro- 
vided we  change  the  sign  before  it. 

We  can  find  the  number  described  by  any  linear  equation  by 
proceeding  in  the  following  way : 

1.  If  any  of  the  numerical  coefficients  are  fractions^  multiply 
both  members  of  the  equation  by  the  least  common  multiple  of  the 
denominators  of  these  coefficients  and  equate  the  products. 

2.  Remove  any  parentheses  that  may  occur  in  the  resulting 
equation. 

3.  Transpose  all  the  terms  that  involve  the  unknown  number  to 
the  left  member,  and  all  the  other  terms  to  the  right  member. 

4.  Simplify  each  member  of  the  resulting  equation  by  com- 
bining like  terms. 

5.  Divide  both  members  of  the  last  equation  by  the  coefficient  of 
the  u7iknown  number  in  the  left  member. 

This  process  gives  a  series  of  equivalent  equations,  and  there- 
fore the  number  given  by  the  last  equation  must  be  the  number 
described  by  the  original  equation. 

It  is  not  necessary  that  these  steps  be  taken  in  this  order. 

34.  Example. 

Solve  the  equation  I^  _  1  =  i£.  (1) 

3        6       4 

Multiplying  each  member  by  12, 

28x-2  =  15a;.  (2) 

Transposing,                 28  x  - 15  a;  =  2.  (3) 

Combining  like  terms,              13  a;  =  2.  (4) 

Dividing  each  member  by  the  coefficient  of  a;, 


42  COLLEGE  ALGEBRA 

This  work  can  be  checked  by  substituting  this  value  of  x  for  x  in  equa- 
tion (1).     The  two  members  of  the  equation  should  then  have  the  same 

value. 

7  •  i^TT     1^14      1  ^  28  -  13  ^  5 

3         6      39      6  78  26 

4  52  ~  26  * 

Since  the  two  members  of  this  equation  have  the  same  value  for  this 
value  of  oj,  we  conclude  that  equation  (1)  has  been  correctly  solved. 

EXERCISES 

Solve  each  of  the  following  equations  or  show  that  it  is  an 
identity,  and  check  your  results : 

1.   2a;+5=5x-f2.  3.   ^^-|(3  +  2x)  =  6. 

•  2.   ^  +  ^  =  ^-0..  4.   ^-30.4-1  =  ^- 

2     3     4  2  2 

5.  (a;  +  l)'  +  (.'c  +  2)2  =  (a;-3)2-}-(aj-4)2. 

6.  (r_l)2  4.(r-2)2=(r-3)2  4-(r-4)2. 

2  4 

8.  a^  4-  63  _  3  rah  =  0.     (Solve  for  r) 

9.  5a;  +  72/  +  3  =  0.     (Solve  for  ic.) 

10.  5  a;  +  T  2/  +  3  =  0.     (Solve  for  2/.) 

11.  aic  +  &2/  +  c  =  0.     (Solve  for  ?/.) 

12.  ax  4-  &2/  +  c  =  0.     (Solve  for  £c.) 

13.  M  =  M  +  pk-     (Solve  for  p^>j 

14.  s  —  So  =  ^ g^i^  —  v4,.     (Solve  for  Wo-) 

15.  2/^  =  2  a («  +  c).     (Solve  for  a;.) 

16.  x^  {y  —  pa?)  =  yp^.     (Solve  for  2/.) 

17.  5-4(a;-l)+|  =  3-4a;. 

18.  p{p-y)=  x{x  +  2/).     (Solve  for  2/.) 


LINEAR  EQUATIONS  IN  ONE  UNKNOWN  43 

19     _^4._^=:44--5I^. 
'   410     330  6765 

^  27 

21 .770     8  .  770 

21.  V  =VQ  +  gt.     (Solve  for  t.) 

22.  a:(m  +  ?i  — 5) +  3 a;  — 8  =  9(7+ a;  — m). 

23.  3(5a;-4)-i(6  +  a')=2. 

25.  23-8=(^-2)(22  +  22  +  4). 

26.  2/  —  2  =  ??i  (x  —  1).     (Solve  for  x.) 

27.  ma5  =  nTF  —  nx.     (Solve  for  x.) 

28.  2.5a;-6.75  =  2.45  +  .75a;. 

29.  From  each  of  the  following  equations  form  a  new  one 
by  equating  the  products  obtained  by  multiplying  each  mem- 
ber by  the  expression  at  the  right.  Do  any  of  the  new  equa- 
tions possess  roots  not  possessed  by  the  corresponding  original 

equations  ? 
^  (a)     a;- 2  =  5.  x-1. 

(h)  3x  +  4  =  7a;-3.  5. 

(c)      a;  +  6  =  0.  X. 

30.  From  each  of  the  following  equations  form  a  new  one 
by  equating  the  quotients  obtained  by  dividing  each  member 
by  the  expression  at  the  right.  Do  any  of  the  new  equations 
possess  all  of  the  roots  of  the  corresponding  original  equations? 


(a)       a;2-l  =  0. 

a;  +  l. 

(6)    x'-lx^O. 

X. 

(c)    2  a; +  10=0. 

2. 

44  COLLEGE  ALGEBRA  \M    N  V' 

PROBLEMS  ^"     V         '^\ 

1.  The  sum  of  three  consecutive  integers  is  24.  What  are 
they  ? 

2.  The  sum  of  three  consecutive  integers  is  a.  What  are 
they  ?  Are  there  any  restrictions  on  a  implied  by  the  condi- 
tions of  this  problem  ? 

3.  The  sum  of  four  consecutive  odd  integers  is  a.  AVhat 
are  they  ?  What  restrictions  are  imposed  on  a  by  the  condi- 
tions of  this  problem  ? 

4.  At  what  time  between  four  and  five  o'clock  are  the 
hands  of  a  watch  opposite  each  other? 

5.  Can  the  hands  of  a  watch  be  opposite  each  other  twice 
within  an  hour  ? 

Count  the  number  of  times  the  hands  of  a  watch  are  oppo- 
site each  other  between  noon  and  midnight. 

6.  How  far  apart  must  the  centers  of  two  pulleys  be  if  each 
has  a  diameter  of  16  inches,  and  the  connecting  belt  is  25  feet 
long? 

Volume  of  an  anchor  ring. — When  the  circle  C  is  revolved  about 
the  line  MN  a  solid  is  generated  that  is  called  an  anchor  ring. 

It  is  shown  in  Integral  Calculus  that 
the  volume  of  this  anchor  ring  is 
equal  to  the  area  of  the  circle  multi- 
plied  by  the*  length  of  the  path  de- 
'      scribed  by  the  center  of  the  circle. 

7.  If  the  radius  of  the  circle  is  3  inches,  how  far  must 
the  center  of  the  circle  be  from  the  axis  in  order  that  the 
volume  of  the  anchor  ring  shall  be  1425  cubic  inches  ? 

8.  What  must  be  the  outer  radius  of  a  circular  ring  that  is 
3  centimeters  wide  and  contains  240  square  centimeters  ? 

9.  What  must  be  the  outer  radius  of  a  circular  ring  that  is 
3  inches  wide  and  contains  a  square  inches  ? 

What  restrictions  on  a  are  implied  in  this  problem  ? 


LINEAR  EQUATIONS  IN  ONE  UNKNOWN  45 

10.  A  runs  around  a  circular  track  in  30  seconds  and  B  in 
35  seconds.  Two  seconds  after  B  starts,  A  starts  from  the 
same  place  in  the  same  direction.     AVhen  will  they  meet  ? 

11.  A  runs  around  a  circular  track  in  a  seconds,  and  B  in 
h  seconds.  Two  seconds  after  B  starts,  A  starts  from  the 
same  place  in  the  same  direction.     When  will  they  meet  ? 

Describe  what  happens  when  a  =  h. 

12.  If  in  Problem  11  A  starts  2  seconds  before  B,  when 
will  they  meet  ? 

13.  If  A  runs  around  a  circular  track  in  40  seconds,  how 
fast  must  B  go  in  order  that  they  may  meet  every  18  seconds 
when  going  in  opposite  directions  ? 

For  the  principles  of  the  lever  involved  in  Problems  14-18  see  the  Ap- 
pendix. 

14.  A  uniform  beam  12  feet  long  weighs  14  pounds  per 
linear  foot,  and  has  a  weight  of  8  pounds  attached  to  it  5  feet 
from  one  end.  When  it  is  lying  horizontal,  how  many  pounds 
must  a  man  lift  in  order  to  pick  up  this  end  of  it  ? 

15.  A  uniform  beam  30  feet  long  balances  on  a  fulcrum  1 
foot  8  inches  from  its  center  when  a  man  weighing  150  pounds 
stands  on  one  end.  How  much  does  the  rail  weigh  per  linear 
foot  ? 

16.  The  roadway  of  a  bridge  is  25  feet  long,  and  the  weight 
of  the  bridge  is  5  tons.  What  pressure  is  borne  by  each  sup- 
port at  the  ends  when  a  wagon  weighing  2  tons  is  one  fourth 
of  the  way  across  ? 

The  bridge  may  be  considered  a  lever  with  the  fulcrum  at  one  of  the 
supports  and  the  power  at  the  other  support.  The  measure  of  the  power 
is  the  reaction  of  the  support  on  the  bridge,  and  is  therefore  equal  to  the 
pressure  on  this  support. 

If  X  =  no.  of  tons  pressure  due  to  the  wagon  on  the  support  nearer  the 
wagon,  then,  neglecting  the  weight  of  the  bridge,  we  have 

25  X  =  2  .  ^. 


Hence, 


^i/^^"^-^^ 


46  COLLEGE  ALGEBRA 

The  pressure  due  to  the  wagon  on  the  other  support  is  then  2  —  x  =  ^. 
But  the  bridge  itself  exerts  a  pressure  of  2^  tons  on  each  support.  Hence 
the  pressures  on  the  supports  are  4  tons  and  3  tons  respectively. 

17.  A  uniform  beam  30  feet  long  and  weighing  175  pounds 
to  the  foot  rests  with  its  ends  on  two  supports.  What  is  the 
pressure  on  each  of  these  supports  when  the  beam  carries  a 
load  of  4  tons  6  feet  from  one  end  ? 

18.  A  beam  is  supported  at  its  ends  and  a  weight  of 
TF  pounds  is  placed  on  it  m  feet  from  one  end  and  n  feet  from 
the  other.  Find  the  pressure  on  each  of  the  supports  due  to 
the  weights. 

19.  A  freight  train  whose  rate  is  18  miles  per  hour  passes 
a  certain  station  h  hours  and  m  minutes  before  a  passenger 
train.  What  must  be  the  rate  of  the  passenger  in  order  that 
it  may  overtake  the  freight  in  t  hours  ? 


CHAPTER   V 

SYSTEMS   OF  LINEAR   EQUATIONS   IN  TWO   OR 
MORE  UNKNOWNS 

35.  We  have  seen  that  a  conditional  equation  of  the  first 
degree  in  one  unknown  is  a  description  of  a  number  and  that 
there  is  only  one  number  that  answers  such  a  description. 
In  this  chapter  we  shall  see  that  an  equation  of  the  first  degree 
in  two  unknowns  is  a  description  of  a  relation  between  two 
numbers,  but  that  this  description  differs  from  the  one  given  by 
an  equation  in  one  unknown  in  that  there  are  a  great  many  num- 
bers related  to  each  other  in  the  way  described  by  the  equation. 

Consider,  for  example,  the  equation 

5x  +  2y  =  4t. 

The  student  can  readily  verify  that  any  pair  of  values  of  x  and  y  in 
the  following  table  are  related  to  each  other  in  the  way  described  by  this 
equation  : 


X 

-8    0 

2 

1 

-3 

-f 

1 

6 

— 

— 

— 

— 

— 

y 

22    2 

-3 

-'i- 

— 

— 

— 

— 

0 

-7 

-1 

-1 

5 

6 

The  student  should  complete  the  table  by  finding  the  values  of  y  that 
are  related  to  the  second  four  values  of  x  in  the  given  way,  and  finding 
the  values  of  x  that  are  related  to  the  last  six  values  of  y  in  the  given  way. 

36.  Graphical  representation.  —  A  description  of  the  rela- 
tion between  two  numbers  by  means  of  an  equation  can  also 
be  given  by  means  of  a  graphical  representation. 

The  following  explanations  and  illustrations  are  given  with 
a  view  to  enabling  the  student  to  become  familiar  with  this 
method  of  representing  these  relations. 

In  Chapter  I  we  looked  upon  the  real  numbers  as  the  repre- 
sentatives of  the  points  of  a  straight  line  with  reference  to  a 

47 


X-L 


•A 

J — J 


48  COLLEGE  ALGEBRA 

given  origin  and  a  given  unit  length.     In  a  similar  way  pairs 
of  real  numbers  can  be  looked  upon  as  the  representatives  of 

the  points  of  a  plane.  We 
take  two  perpendicular 
straight  lines  in  this  plane 
and  their  point  of  intersec- 
tion as  the  origin  on  each 
line.    If  X  and  y  are  a  pair 

-• X       of  numbers  we  lay  off  x  on 

the  horizontal  line,  in  the 
way  just  recalled,  and  y  in 
a  similar  way  on  the  verti- 
cal line,  counting  distances 
above  the  origin  as  positive 
and  those  below  as  nega- 
tive. Then  through  the  point  representing  each  of  these  num- 
bers we  draw  a  line  perpendicular  to  the  line  on  which  the 
point  lies.  The  point  of  intersection  of  these  two  perpen- 
diculars is  the  point  represented  by  the  pair  of  numbers  x 
and  y. 

In  the  figure  given  above,  the  points  A,  B,  C,  and  D  are  represented 
by  the  pairs  of  numbers  (2,  1)  ;  (-  f,  5)  ;  (- 1,  —  |)  ;  and  (2,  -  -J), 
respectively.  We  can  also  say  that  these  four  points  represent  these 
four  pairs  of  numbers  respectively. 

37.  Definitions.  —  If  the  numbers  x  and  y  represent  the 
point  A  in  the  way  just  described,  x  is  called  the  abscissa  of 
Aj  and  y  its  ordinate.  The  two  numbers  together  are  called 
the  coordinates  of  the  point. 

The  abscissa  of  a  point  is  always  given  first.  Thus,  the 
point  (2,  3)  is  the  point  whose  abscissa  is  2  and  ordinate  3. 

The  two  perpendicular  lines  are  called  the  axes  of  coordi- 
nates, or  coordinate  axes. 

The  axis  on  which  the  abscissae  are  laid  off  is  called  the 
X-axis,  and  the  one  on  which  the  ordinates  are  laid  off  is  called 
the  j^-axis.     The  ic-axis  is  usually  horizontal. 


EQUATIONS  IN  TWO  OR  MORE   UNKNOWNS       49 

38.  Every  pair  of  values  of  x  and  y  is  represented  by  a 
point,  and  every  point  represents  a  pair  of  values  of  x  and  ?/, 
which  are  its  coordinates.  These  can  be  found  by  drawing 
perpendiculars  from  the  point  to  the  axes  and  measuring  the 
distances  from  the  origin  to  the  feet  of  these  perpendiculars. 


f),  (1,  1),  (4,  4). 


4). 


EXERCISES 

Draw  two  axes  and  locate  the  following  points,  using  \  inch 
or  1  centimeter  as  the  unit  distance : 

1.  (2,3),  (5,  -2),  (0,1). 

2.  (-2, -2),  (-3,-3),  (- 
3-   (-f,|),(0,0),  (3, -3),  (i, -^; 

4.  (7,  0),  (5,  0),  (4,  0),  (1,  0),  (-3,  0),  (-  8,  0). 

5.  (0,  -10),  (0,  -  8),  (0,  -  7),  (0,  -4),  (0, 1),  (0,  2),  (0,  3). 

6.  Determine  by  accu- 
rate measurements  the 
coordinates  of  the  points 
A,  B,  C,  D,  E,  and  F  in 
the  accompanying  figure. 

7.  Draw  a  line  parallel        ^ 
to  the  a;-axis  and  3  units 
above  it,  and  measure  the 
coordinates    of    5    points 
on  it. 

8.  Draw  a  line  parallel 
to  the  2/-axis  and  2  units 
to  the  left  of  it,  and  measure  the  coordinates  of  5  points  on  it. 

9.  Lay  off  the  points  whose  coordinates  are  the  successive 
pairs  of  values  of  x  and  y  in  the  table  of  §  35.  Can  a  straight 
line  be  drawn  through  these  points  ? 


?Z) 


50 


COLLEGE  ALGEBRA 


x^ 


39.  Locus  of  an  equation.  — For  a  given  equation  in  x  and  y 
there  is  a  certain  line,  straight  or  curved,  such  that  the  coordi- 
nates of  every  point  on  the  line  satisfy  the  equation,  and 
that  every  pair  of  values  of  x  and  y  that  satisfy  the  equation 
are  the  coordinates  of  a  point  on  the  line. 

This  line  is  called  the  locus 
of  the  equation.  It  exhibits 
graphically  the  relation  between 
the  unknowns  (or  variables) 
that  is  described  by  the  equa- 
tion in  the  sense  that  those 
numbers  which  are  the  coordi- 
nates of  points  of  this  line,  and 
only  those  numbers,  are  related  to  each  other  in  the  way 
described  by  the  equation. 

For  example,  the  locus  of  the  equation 
2  X  -  3  .V  =  6 
is  the  straight  line  of  the  accompanying  figure. 


EXERCISES 

Lay  off  carefully  10  points  on  the  locus  of  each  of  the 
following  equations : 

1.  x  =  y. 

2.  x  —  y  =  5. 

3.  2x  =  3y. 


5.  -5»+62/+7  =  0. 

6.  a;  +  2/  +  l  =  0. 

7.  x  =  0. 


8.  y  =  0. 


9. 

x  +  y  =  0. 

14. 

-^-•^  =  1. 
5      8 

10. 

y  =  Sx. 

15. 

4a; -f  5  =  0. 

11. 

M=^- 

16. 
17. 

a;-4  =  0. 
2/  +  5  =  0. 

12. 

-M=^- 

18. 

Sx-^5y  =  2. 

13. 

i-.=i. 

19. 
20. 

y-l  =  3{x-2). 
y+S  =  -5(x-l) 

EQUATIONS  IN  TWO   OR  MORE  UNKNOWNS       51 

40.  Graphical  solution.  —  It  is  proved  in  Analytic  Geometry 
that  the  locus  of  every  equation  of  the  first  degree  in  two  variables 
is  a  straight  line. 

It  is  because  of  this  fact  that  equations  of  the  first  degree 
are  called  linear  equations. 

Any  pair  of  values  of  the  unknowns  that  satisfy  each  of  two 
linear  equations  in  two  unknowns  must  be  the  coordinates  of 
a  point  on  the  locus  of  each  equation ;  that  is,  must  be  the 
coordinates  of  the  point  of  intersection  of  these  loci.  Hence, 
since  two  straight  lines  intersect  in  not  more  than  one  point, 
two  linear  equations  in  two  unknowns  cannot  be  satisfied  by 
more  than  one  pair  of  values  of  the  unknowns  unless  they  are 
dependent.  (See  §  42.)  Thus  we  see  that  although  a  single 
linear  equation  in  two  unknowns  does  not  describe  two  par- 
ticular numbers,  two  such  equations  taken  together  do  give  us 
such  a  description,  provided  that  they  are  not  dependent  and 
not  inconsistent.    (See  §  42.) 

The  values  of  the  unknowns  that  satisfy  two  linear  equa- 
tions can  be  found  by  plotting  the  loci  of  these  equations  and 
measuring  the  coordinates  of  their  point  of  intersection ;  but 
results  obtained  in  this  way  are  only  approximate  since  the 
locus  of  an  equation  cannot  be  drawn  with  perfect  accuracy 
and  since  the  coordinates  of  a  point  of  intersection  of  two  loci 
cannot  be  measured  with  perfect  accuracy. 

41.  Algebraic  solution.  —  These  values  of  the  unknowns 
can  also  be  found  without  reference  to  the  loci  of  the 
equations. 

We  shall  illustrate  two  ways  of  doing  this : 

1.   Solution  by  the  method  of  substitution. 

rind  the  numbers  described  by  the  equations 

3a;-4y^7,  (1) 

7x  +  5y=9.  (2) 

Solve  (1)  for  x,  x  =  '^-^^ .  (3) 

3 


52  COLLEGE  ALGEBRA 

Substitute  this  value  of  x  for  x  in  (2), 


3 

Solve  (4)  for  y,           49  +  28  ?/  +  15  ?/  =  27, 

43  2/ =-22, 

^         43 
Substitute  this  value  of  ?/,  for  ?/  in  (3),          gg 

"^"43 
^=      ^      = 

_71 
43 

(4) 


Check.  —  From  (1), 

3.71-4f_22U§01  =  7. 
43         \     43;       43 

From  (2),  7  •  ^  +  5  f  -  ^^^  =  §^  =  9. 

^  ^'  43         V      43;       43 

From  a  study  of  this  example,  the  student  should  formulate  a  general 
rule  of  procedure. 

2.    Solution  by  the  method  of  addition  or  subtraction. 

Multiply  each  member  of  equation  (1)  by  5  and  each  member  of  (2) 

by  4. 

15a;-202/  =  35.  (6) 

28  a;  +  20  ?/  =  36.  (6) 

The  sum  of  the  left  members  of  (5)  and  (6)  equals  the  sum  of  the 
right  members.     Hence,  _ 

"^-43- 
Substitute  this  value  of  a;  for  x  in  (1), 

4b  ^ 

213  -  172  ?/  =  301, 
172  2/ =-88, 

The  check  is  the  same  as  in  the  other  method. 

From  a  study  of  this  example  the  student  should  formulate  a  general 
rule  of  procedure. 


EQUATIONS  IN  TWO  OR  MORE  UNKNOWNS       53 

42.  Inconsistent  and  dependent  equations.  —  Sometimes  two 
linear  equations  in  two  unknowns  have  no  common  solution. 
This  is  the  case  when  the  loci  of  the  two  equations  are  parallel 
lines.     The  student  should  verify  that 

3  aj  _|_  4  2/  =  1, 

and  S X  -^  iy  =  5 

are  two  such  equations. 

Such  equations  are  said  to  be  inconsistent. 

Sometimes  two  equations  convey  the  same  information  con- 
cerning the  relations  of  the  unknowns.  Such  equations  are 
said  to  be  dependent.  Two  dependent  equations  have  the 
same  locus. 

The  equations  3  a;  -j-  4  2/  =  1, 

and  6x-{-8y  =  2 

are  dependent  equations.     The  second  one  gives  no  information 
concerning  x  and  y  that  is  not  given  by  the  first  one. 

EXERCISES 

Solve  the  following  pairs  of  equations  graphically  and  also 
by  one  of  the  other  methods.  See  that  the  results  agree 
approximately. 

1.  2x-y  =  l, 
3x-\-2y  =  12. 

2.  x-^y-\-S  =  0, 
4,x-5y +  3  =  0. 

3.  5x  +  2y=2, 
7x-5y  =  S4:.  7.    2x-Sy-\-4t  =  0, 


5. 

X 

3 

-y=- 

5 

-1, 

4^2 

=  1. 

a 

3; 

x+4:y 

+  5 

=  0, 

6; 

x+8y 

=  — 

10. 

3y-2x-2=0. 

3-4:X  =  y, 
2  '  4    .   '  Sx  +  2y  =  6, 


4.   2x-^3y  =  6y 

«_l_^^l^  8.   3-4:X  =  y, 


54  COLLEGE  ALGEBRA 

x  —  5 


y-4:  = 


-8" 


12. 

3x-\-4.y-l=z 

0, 

12x  —  5y-^6  = 

=  0. 

13. 

5x-2y  +  ll  = 

=  0, 

e-i.=y^. 

14. 

a     b 

ax  -\-by  =  c. 

10.  ?i±i_^jz^ 

6  -9  '. 

11.  x  +  y  +  l  =  (), 

The  lines  of  Ex.  14  cannot  be  plotted  until  definite  values 
are  assigned  to  a,  h,  and  c. 

17.  ^+y=i, 

m     n 

_E ^  =  1. 

4  m      5  n 

In  solving  the  following  equations  do  not  clear  of  fractions, 

but  solve  for  -  and  -.    These  equations  are  not  linear  and  their 
X  y 

loci  are  not  straight  lines.    Do  not  try  to  solve  them  graphically. 


1.    ax  -\-  by  +  c  =  0, 

16.    ^  +  1  =  1, 

Ix  4-  my  +  76  =  0. 

a     b 

^-^  =  1 

a     b 

18. 

X     y 

19. 

r   9    „ 

20.   «  +  ^  =  e, 

X     y 

12+4  =  1. 
a;      2^ 

-  +  -  =/• 
a;      2/ 

43.    Solution  by  formulae.  —  Consider  the  two  equations 

a^x  +  hy  =  Ci,  (1) 

ttgic  -{-  b^  =  Cg.  (2) 


The  student  should  verify  that  if  «i&2  —  «2&i  ^^  0  the  numbers  described 

C162  —  C2&1 

0162  —  0,2^1 


by  these  equations  are  h   —  r  h 


and  2/  =  ^^"^  -  ^^^^ 

ai&2  —  Cf20l 


EQUATIONS  IN  TWO  OR  MORE   UNKNOWNS       55 

These  formulae  can  be  easily  remembered  by  means  of  the 
following  device : 

The  symbol   ^'    ^' 
the  cross  products  0^)2  and  a^b^.    That  is, 


will  be  used  to  represent  the  difference  of 
CI2    62 


tti^a  —  0t2^i' 


For  example, 

12    51 
U    3 


6  _  20  =  -  14,  and 


=  i-(-56)  =  56i. 


This  symbol  is  called  a  determinant  of  the  second  order. 
The  values  of  x  and  y  which  were  obtained  as  the  solution 
of  the  preceding  equations  can  now  be  expressed  as  follows : 


X  = 


Cl  61 

ai    Cl 

C2   62 

,  y  = 

a^    c-z 

ai  bi 

«i  &i 

a2    62 

a2    h2 

44.  We  observe  that  the  two  denominators  are  the  same 
determinant.  The  numbers  in  the  first  row  of  this  deter- 
minant are  the  coefficients  of  x  and  y  respectively  in  the 
first  equation,  and  the  numbers  in  the  second  row  are  the 
coefficients  of  x  and  y  in  the  second  equation.  The  numer- 
ator of  the  fraction  that  gives  the  value  of  a;  is  the  same 
as  the  denominator  with  the  exception  that  the  coeffi- 
cients of  X  (the  numbers  in  the  first  column)  have  been 
replaced  by  the  constant  terms  of  the  respective  equa^ 
tions ;  and  in  the  fraction  that  gives  the  value  of  y  the 
numerator  is  the  same  as  the  denominator  with  the  exception 
that  the  coefficients  of  y  (the  numbers  in  the  second  column) 
have  been  replaced  by  the  constant  terms  of  the  respective 
equations. 

By  means  of  these  formulae  we  can  write  down  immediately 
the  solutions  of  any  pair  of  consistent,  independent,  linear 
equations  in  two  unknowns. 


56  COLLEGE   ALGEBRA 

For  example,  the  values  of  x  and  y  that  satisfy  the  equations 
7x+  12?/  =  3 
and  a;  -  2  w  =  5 


X  = 


and 


3 

12 

6 

-2 

_  -6-00 
-  14  -  12 

33 

7 

12 

"13' 

1 

-2 

7 

3 

1 

5 

85-    3 
-  14  -  12 

16 

7 

12 

13 

1 

-2 

The  student  should  observe  that  these  formulae  apply  only 
when  each  equation  is  in  the  form  ax  -\- hy  =  c.  If  it  is 
not  in  this  form  originally,  it  must  first  be  reduced  to  this 
form.  If  one  of  the  letters  does  not  appear  in  an  equation, 
it  should  be  regarded  as  present  with  the  coefficient  zero. 
(See  Ex.  11.) 

In  solving  Equations  (1)  and  (2)  of  §  43  we  get 


and 


{afii  —  ajb^)x  =  C162  —  C261 


The  common  coefficient  of  x  and  y  in  these  equations  is  the  de- 
terminant in  the  denominators  of  the  formulae,  while  the  right 
members  of  the  equations  are  the  determinants  in  the  numera- 
tors of  the  formulae. 

Hence  if  in  these  formulae  the  determinant  in  the  denomina- 
tors is  zero,  while  those  in  the  numerators  are  different  from 
zero,  the  equations  are  inconsistent.  If  all  the  determinants 
are  zero,  the  equations  are  dependent. 


EQUATIONS  IN  TWO  OR  MORE   UNKNOWNS       57 

EXERCISES 

Solve   the   following   pairs   of   equations  by  means  of  the 
formulae : 

10.  Sx  —  5y  =  2, 

11.  5  05 +  19  2/ =  14, 
3  a;  =  2. 

12.  kx  -\-  ly  -\-  m  =  Of 
2x  —  Sy  =  4:m. 

13.  a(x  —  y)-\-b(x-\-y)=b  +  c, 
a{x  -\-y)—  b(x  —  y)=:b  —  c. 

14.  ^-  +  f  =  1, 
a      b 

^  +  ^  =  1. 
2a      46 

15.  (a-\-b)x-\-(a  —  b)y  =  Sab, 
(a  —  b)x  +  (a  -f  b)y  —  ab. 

16.  ax  -{-  by  -\-  c  =  Of 
ax  —  by  —  c  =  0. 

17.  Ix  +  my  =  n, 
2  Za;  +  3  my  =  n. 

18.  3  «  +  1  =  5, 

x-\-y  =  2. 

19.  a(aj  +  2/)  +  6(aJ  -  y)  =  a, 
(a  4-  6)a?  +  (a  —  6)?/  =  6. 

c      a 


1. 

9x-\-y  =  5, 

3x-\-ny  =  ll. 

2. 

5  +  7"^^ 

^  -  -^  =  1. 
3      4 

3. 

2/  -  5  =  4  a;, 

2/-3=:7(a:  +  l). 

4. 

.v-|=H^  +  i). 

y-i  =  H^-i)' 

5. 

y-T      a;  +  10 
5     "-      3     ' 

a;  _  4      y-S 

2^3* 

6. 

4a54-3y  =  2, 

8aj  +  6?/  =  l. 

7. 

5^2        ' 

3^7 

8. 

-  +  ^  =  1, 
4^2 

a;  +  2  2/  =  4. 

9, 

a;  _  1  _  ^  _  2 
3            -8  ' 

.  +  1=1. 

58  COLLEGE  ALGEBRA 

45.  Two  descriptions  of  the  relations  connecting  three  un- 
known numbers  are  not  sufficient  to  identify  these  numbers. 

Eor  example,  there  are  many  sets  of  values  of  x,  y,  and  z 
that  answer  the  two  following  descriptions : 

2a;+32/  +  2  =  4,  (1) 

-4a;4-5y  +  62;  =  7.  (2) 

The  student  should  make  a  table  exhibiting  at  least  five  sets  of  values 
of  X,  y,  and  z  that  answer  these  descriptions.  How  many  more  such  sets 
of  values  are  there  ? 

There  is  however  only  one  set  of  values  of  the  unknowns 
that  answer  three  descriptions  of  the  first  degree  of  the  rela- 
tions connecting  them.     Consider,  for  example,  the  description 

x+y+z^Z,  (3) 

in  connection  with  (1)  and  (2). 

If  we  multiply  each  member  of  (1)  by  2  and  add  each  mem- 
ber of  the  resulting  equation  to  the  corresponding  member  of 
(2),  we  get  11 2, +  8.  =  15.  (4) 

Then  if  we  multiply  each  member  of  (3)  by  2  and  subtract 
the  products  from  (1),  we  get 

y-z  =  -2,  (5) 

Now  every  set  of  values  of  x,  y,  and  z  that  satisfy  equations 
(1),  (2),  and  (3)  must  also  satisfy  equations  (4)  and  (5).  But 
the  only  values  of  y  and  z  that  satisfy  (4)  and  (5)  are 

Moreover  if  y  and  z  have  these  values,  we  must  have  a?  =  -f^ 
in  order  that  (1),  (2),  and  (3)  be  satisfied.  Hence  there  is  only 
one  set  of  values  of  a?,  y,  and  z  that  satisfy  equations  (1),  (2), 
and  (3). 

It  was  not  necessary  to  proceed  in  exactly  this  way  in  order 
to  get  the  solution  of  the  given  equations.     We  might  have 


EQUATIONS  m  TWO  OR  MORE  UNKNOWNS       59 

chosen  a  different  unknown  for  the  first  elimination  and  ac- 
cordingly have  combined  the  original  equations  differently. 
But  the  same  variable  must  always  be  eliminated  in  the  first 
two  eliminations. 

The  work  of  solving  a  system  of  three  linear  equations  in 
three  unknowns  can  be  checked,  as  in  the  case  of  equations 
with  two  unknowns,  by  substituting  in  the  equations  the 
values  found  for  the  unknowns. 

46.  Inconsistent  and  dependent  equations.  —  It  may  be  that 
there  are  no  values  of  ic,  y^  and  z  that  satisfy  all  three  equa- 
tions. 

Consider,  for  example,  the  following  equations  : 

2x+3y+5r  =  4,  (1) 

x-\.y  +  z  =  ^,  (2) 

4  X  +  6  y  +  2  0  =  5.  (3) 

If  we  multiply  each  member  of  (1)  by  2  and  subtract  the  products 
from  the  corresponding  members  of  (3),  we  get 

0  =  -3. 

This  obvious  untruth  was  obtained  on  the  supposition  that  there  is  a 
set  of  values  of  cc,  y^  and  z  that  satisfy  both  (1)  and  (3).  Hence  the  sup- 
position must  be  wrong  and  (1)  and  (3)  are  inconsistent. 

Exercise.  —  Explain  in  detail  where  the  existence  of  a  set  of  values  of 
oj,  y,  and  z  satisfying  (1)  and  (3)  was  assumed  in  getting  the  result  0  =—  3. 

Sometimes  the  three  equations  are  such  that  one  of  them 
gives  no  more  information  concerning  the  relations  connecting 
the  unknowns  than  is  given  by  the  other  two  equations. 

The  equations  2  x  -|-  3  y  +  4  2;  =  5,  (1) 

-  «  +  2  y  -  3  ;?  =  2,  (2) 

x^hy  +  z  =  'J,  (3) 

form  such  a  set,  inasmuch  as  the  two  members  of  (3)  are  the  sums  of  the 
corresponding  members  of  (1)  and  (2),  and  any  set  of  values  of  a:,  y,  and 
z  satisfying  (1)  and  (2)  must  also  satisfy  (3). 

The  student  should  give  to  z  in  (1)  and  (2)  any  convenient  value  and 
solve  the  resulting  equations  for  x  and  y.  He  will  then  find  that  these 
values  of  a:,  y,  and  z  satisfy  (3) . 


60  COLLEGE  ALGEBRA 

Such  equations  are  said  to  be  dependent.  If  each  of  the 
equations  contains  information  not  contained  in  the  others, 
the  equations  are  said  to  be  independent. 

Note, — If  the  student  has  studied  solid  geometry,  he  maybe  inter- 
ested in  the  fact  that  an  equation  of  the  first  degree  in  three  variables 
represents  a  plane,  just  as  an  equation  of  the  first  degree  in  two  variables 
represents  a  straight  line.  The  loci  of  two  inconsistent  equations  are 
parallel  planes.  The  loci  of  three  dependent  equations  are  three  planes 
through  the  same  line. 

EXERCISES 
'    Solve  the  following  sets  of  equations  and  check  your  results : 


1. 

2x-^Sy-z  =  5, 

8. 

A.B.O     o 

3  +  2+^-^' 

2. 

4:X  +  2y  +  2z  =  U. 
2x-4:y-6z  =  -6, 

2^5-^3-^' 

Sx-\-5y-\-2z=lS, 

—  x-{-2y-\-z  =  0. 

ABC     63 

5       3     2       5  * 

3. 

Tx-\-5y-\-Sz  =  S, 

1     1      , 

—  6x-{-2y-^oz  =  6j 

9. 

X      y 

14  a;  +  10  2/  +  16  ^  =  4. 

4. 

a;  +  2/  +  l  =  0, 

y     z 

y  +  z  =  4, 

1      1 

z-{-x=l. 

Z        X 

6. 

-x-y  +  2z=:12, 
4:X-Sy-2z  =  10, 

Solve  for  -,  -,  and  i 
X    y           z 

2x  +  4:y  +  Sz  =  3. 

6. 

ax-^by  =  1, 
cy-\-dz  =  1, 
ez-\-fx  =  l. 

10. 

^  +  B+0  =  3, 
A-\-B-C=2, 
A-B+C=5. 

7. 

4.A-5B  +  6C=23, 

11. 

A-{-B==0, 

2A-\-7  B-{-3C=A0, 

B+C=l, 

9^  +  2^  +  8(7=61. 

C  +  A  =  3. 

EQUATIONS  IN  TWO  OR  MORE   UNKNOWNS       61 

12.  14^  +  20/+c  =  -149,  17.    3^1  +  2  5  +  4  0+4=0, 
16  g  +  18/+  c  =  - 145,  2A-5B-G  =  20, 
65^  +  4/- c  =  13.  ^  +  5- (7=0. 

13.  c  =  0,  18.    Ax-3y-4.z  =  0, 
2/+c  +  l  =  0,  5a;  +  62^  +  102;  =  7, 
2gr  +  c  +  l  =  0.                              x-\-2z=^l. 

14.  5  +  2^  +  4/+c  =  0,  ^^-   12x  +  5y  +  Sz^-U, 
5  +  4^  +  2/+ 0=0,  Ta;  +  2/  +  2;+|  =  0, 


15. 


16. 


13+6^-4/+c  =  0.  ^  +  2^  +  2.  =  -3. 

20. 


X 

+  2/  +  2;=3, 

X 

-2y  +  z=:0, 

3 

x-\-4:y  —  z  =  6. 

X 

2 

+  !  +  .  =  !, 

X 

-M=^' 

- 

■i-^+l=^- 

tt     6      c 

a     0     c 

a     0     c 

-1. 

aiX-\-biy  +  c^z  = 

=  ^1, 

a^  +60^  +  ^22:  = 

=  ^2, 

«3aJ  +  &32/  +  C32;  = 

=  t^3. 

21. 


47.  Solution  by  formulae.  — The  equations  in  Ex.  21  of  the 
preceding  article  are  typical  of  every  set  of  three  simultaneous 
linear  equations  in  three  unknowns,  since  any  such  set  can  be 
obtained  from  these  by  assigning  proper  values  to  ai,  b^  Ci,  c?i, 
<^2>  ^2?  <hf  ^2y  ^3}  ^3?  ^3?  3-11^  ^3-  Hcuce  the  values  of  x,  y,  and  z 
that  satisfy  these  equations  constitute  formulae  for  the  solu- 
tion of  any  set  of  three  linear  equations  in  three  unknowns. 

If  016263  +  «2&3Ci  +  asbiCi  —  azboC\  —  a\hzc<2.  —  aob\Cz  ^  0  these  values 
of  x,  y,  and  z  are, 

dih^Cz  +  O/ihzCi  +  a3&iC2  —  0362^1  —  a\bzc<>  —  026103 
, .  _  did^cz  +  g2<?3<?i  +  03<^iC2  —  azd2C\  —  a\d%C2  —  02^1^3 

016263  +  «263Cl  +  036102  —  036261  —   O163C2  —  O261C3 
_  Oi62<^3  +  0263<?1  +  036l(?2  —  azh'2.d\  —  Oi63d?2  —  0261(^3  ^ 

0162C3  +  026301  +  036102  —  O36201  —  O16302  —  O26103 


62 


COLLEGE  ALGEBRA 


These  formulae  can  be  remembered  easily  by  means  of  the 
following  device : 

will  be  used  to  represent  the  ex- 


The  symbol 


pression  ai&aCa 


1 

'2        ^2 


The  six  terms  of  this  expression  are  formed  in  the  following 
way :  the  first  and  second  columns  of  the  symbol  just  de- 
scribed are  rewritten  at  the  right  of  the  symbol  thus : 


Then  the  products  of  the  numbers  in  the  diagonals  running  down 
from  left  to  right  are  formed,  and  also  of  the  numbers  in  the  diag- 
onals running  down  from  right  to  left.  The  signs  of  these  last 
three  products  are  changed.  The  algebraic  sum  of  these  six  prod- 
ucts with  their  signs  thus  modified  is  the  value  of  the  symbol. 

For  example, 

2    3        1 

2.2.7  +  3(-6).l  +  1.5.4-1.2.1-2(-6)  .4-3.5.7 
=  28-18  +  20-2  +  48-105 
=  -  29. 


2     -6 
4         7 


This  symbol  is  called  a  determinant  of  the  third  order. 
The  values  of  x,  y,  and  z  that  were  obtained  as  the  solution 
of  the  given  equations  can  now  be  expressed  as  follows : 


di    61    ci 

ai    dx    ci 

ai    61    c?i 

d2    62    C2 

a^    d-2    C2 

a2    &2    di 

dz    63    C3 

,   y  = 

as    dz    C3 

1    «  = 

as    6s    dz 

ai    61    Ci 

ai    61    Ci 

ai    hi     Ci 

0,2      62      Co 

a2    &2    C2 

a2    62    C2 

as    h    C3 

as    63    C3 

as    &s     C3 

48.   We  observe  that  the  three  denominators  are  the  same 
determinant.     The  numbers  in  the  first  row  of  this  determinant 


EQUATIONS  IN  TWO  OR  MORE  UNKNOWNS       63 


are  the  coefficients  of  x,  y,  and  z  respectively  in  the  first  equation, 
and  the  numbers  in  the  second  and  third  rows  are  the  respective 
coefficients  of  these  unknowns  in  the  second  and  third  equations 
respectively.  The  numerator  of  the  fraction  that  gives  the  value 
of  X  is  the  same  as  the  denominator  with  the  exception  that  the 
coefficients  of  x  (the  numbers  in  the  first  column)  have  been 
replaced  by  the  constant  terms  of  the  respective  equations.  The 
numerators  of  the  fractions  giving  the  values  of  y  and  z  are 
related  to  their  respective  denominators  in  an  analogous  way. 

49.    Suppose  we  wish  to  solve  the  equations 

x-iry-\-z-l  =  0, 
x  —  y  —  z  =  1, 
2x  +  Sy  =  S, 
by  means  of  these  formulae. 

We  observe  that  the  first  equation  is  not  in  the  form  assumed  in  the 
preceding  discussion  —  the  constant  term  must  be  in  the  right  member. 
Hence  we  rewrite  the  equations  as  follows  : 

x  +  y  +  z  =  l, 
x-y  -z  =  l, 


Then,  by  the  formulae, 


2x-hSy  =  S. 


fr  — 

8         3 

_3-8  +  8+3_  J 

3_2+2+3 

y  = 

-1 

8-2-24-8      o 

1-1 

-1 

-            6                -' 

1     -1 

__8 +3+2+2-3-8 

6 

2         3 

0 

=  -2. 


64  COLLEGE  ALGEBRA 

50.  It  can  be  proved  that  if  the  determinant  in  the  denomi- 
nators in  the  formulae  at  the  end  of  §  47  is  zero  and  the  de- 
terminants in  the  numerators  are  not  all  zero,  the  equations 
are  inconsistent. 

Consider,  for  example,  the  equations 

x  +  y-2zz=l, 

6x  +  6y  -\-z  =  9. 

It  can  be  shown  that  the  equations  may  also  be  inconsistent 
when  all  the  determinants  of  these  formulae  are  zero.  The 
situation  here  differs  in  this  respect  from  what  we  found  in  the 
case  of  two  linear  equations  in  two  unknowns.     (§  44.) 

Consider,  for  example,  the  equations 

3x4-  4y-20  =  l, 
6x+  8^  —  40  =  4, 
9x+ 12  y-6z  =  5. 

EXERCISES 
Solve  the  following  sets  of  equations  by  the  formulae : 

1.  x  +  2y-^z  =  0,  4.  ^4-0=1, 
x-2y-S  =  0,  B+A=3, 
x-\-y-}-z  =  S.  2C-B  =  1. 

2.  x-\-y-\-2z  =  0,  5.   2x  —  Sy-{-4:Z  =  l, 
2x-y-2z-l  =  0,  x-\-y-5z  =  2, 
Sx-\-6z-5y  +  l  =  0.                   3x-2y-z  =  3. 

3.  A-{-B+C=3,  6.  2l-Sn  +  m  =  2, 
5^  +  454-3  0=5,  Z4-w4-^i  +  l  =  0, 
6A  +  SB  +  2C=2.  Z-4m-4n  =  5. 

7.    -4^-554-0=7, 
2^4-3(7=10, 
-55  4-70  =  3. 


EQUATIONS  IN  TWO  OR  MORE  UNKNOWNS       65 


8.  A-{-B  =  2, 
^-f  C  =  4, 
C  +  .4  =  3. 

9.  x-y-{-z  =  2, 
y-z  =  -S, 

z  —  x  =  6. 


-  M-I=^' 


b      Z 


4^3^10 


51.  Geometrical  applications.  —  The  discussion  of  geometri- 
cal principles  given  in  this  article  and  the  following  one  leads 
up  to  an  important  geomet- 
rical use  of  systems  of  three 
linear  equations  in  three 
unknowns. 

Denote  the  two  points 
(2,  4)  and  (5,  2)  by  P  and 
Q  respectively.  Draw  PM 
and  QN  parallel  to  the  y-axis 
and  QR  parallel  to  the  avaxis. 
Connect  P  and  Q. 

Now,   EQ  =  MN  =  5-2,   since  0M=2   and  ON  =5-,  and 
EP  =  4.-2,  since  MP=  4  and  MR  =  2.     Hence, 


y 

1 

R 

> 

Q 

V 

0 

A 

/                           N 

PQ  =  V(5  -  2)2  +  (4  -  2)2  =  V13. 

In  general,  if  P  represents  the  point  (x^,  y^)  and  Q  the  point 
(^2j  2/2)?  RQ  =  X2  —  Xi  and  RP  =  2/1  —  y^-     Hence, 


PQ  =  ^(x,  -  a^O'  +  (2/1  -  2/2)=  =  V  {X,  -  x,y  +  (2/2  -  2/1)'. 
If  we  represent  the  distance  PQ  by  d,  we  have  the  formula 

This  important  formula  enables  us  to  find  the  distance  be- 
tween two  points  when  the  coordinates  of  these  points  are 
given. 


66  COLLEGE  ALGEBRA 

EXERCISES 
Find  the  distances  between  the  following  pairs  of  points : 

1.  (1,3),  (2,5).  3.   (-2,  -7),  (-3,6). 

2.  (0,0),  (4, -3).  4.   (10,  0),  (0,  2). 
6.   (8, -3),  (-3,8). 

6.  Show  that  the  quadrilateral  whose  vertices  are  the  points 
(^>  2),  (—4,  2),  (—4,  —5),  and  (3,  —5)  respectively  has  equal 
sides. 

52.  The  equation  of  a  circle.  —  Every  point  in  the  circum- 
ference of  a  circle  whose  radius  is  5  and  whose  center  is  at  the 
point  (2,  3)  is  5  units  distant  from  the  center. 

Hence  if  x  smd  y  are  the  coordinates  of  such  a  point,  we  have 

(x-2y  +  (y-Sy  =  25. 

If  X  and  y  are  the  coordinates  of  a  point  inside  the  circle, 
^^^"^  (a^-2)2  +  (2/-3)2<25; 

and  if  they  are  the  coordinates  of  a  point  outside  the  circle, 
^^^"^  (x-2y  +  {y-Sy>25. 

The  points  in  the  circumference  of  this  circle  are  therefore 
the  only  points  whose  coordinates  answer  the  description  given 
by  this  equation.  We  say  accordingly  that  this  equation  is 
the  equation  of  the  circle  and  that  the  circle  is  the  locus  of 
this  equation. 

EXERCISES 

Write  the  equations  of  the  following  circles,  and  draw  the 

circles : 

1.  Radius  3  and  center  at  (—2,  1). 

2.  Radius  1  and  center  at  (0,  0). 

3.  Radius  4  and  center  at  (4,  0). 

4.  Radius  6  and  center  at  (0,  6). 


EQUATIONS  IN  TWO  OR  MORE  UNKNOWNS       67 

5.  Radius  2  and  center  at  (2,  2). 

6.  Radius  5  and  center  at  (—  3,  —  4). 

7.  Radius  3  and  center  at  (5,  —  2). 

8.  Radius  4  and  center  at  (0,  0). 

9.  Radius  7  and  center  at  (5,  3). 
10.   Radius  3|-  and  center  at  (7,  —  6). 

53.   The  equation  (x  -  hf  +  (y-  ky  =  r^, 

which  is  the  equation  of  the  circle  with  the  radius  r  and  the  . 
center  at  the  point  {h,  k),  can  be  written  in  the  form 

ic2  +  2/'  -  2  /ix  -  2  %  +  7i2  -f  A;2  -  ?^  =  0. 

This  is  equivalent  to  the  form 

x^  +  y'  +  2gx  +  2fy-\-c  =  0, 

if  we  let  2  gr  stand  for  —  2  /i,  2/  stand  for  —2  k,  and  c  stand 
for  h^-\-k''-  r". 

The  equation  of  every  circle  can  be  written  in  this  form. 

We  can  take  advantage  of  this  fact  to  find  the  equation  of 
the  circle  that  passes  through  three  given  points  not  on  the 
same  straight  line. 

For  example,  suppose  that  we  want  to  find  the  equation  of  the  circle 
that  passes  through  the  points  (7,  10),  (8,  9),  and  (—3,  —  2). 
The  equation  must  be  in  the  form 

x'^-\-y2  +  2gx-\-2fy  +  c  =  0, 

and  it  must  be  satisfied  by  the  coordinates  of  each  of  these  points.    Hence, 

49  +  100  +  14^^  +  20/+  c  =  0, 
64  +  81  +  16  gr +  18/4-0  =  0, 
9  +  4-6sr-4/+c  =  0. 
Or  14  5r  +  20/+c=-149, 

16p  +  18/+c=-145, 
65r  +  4/- c  =  13. 


68  COLLEGE  ALGEBRA 

This  is  a  system  of  three  linear  equations  in  the  three  unknowns, 
gr,  /,  and  c.  If  we  solve  these  equations  by  either  of  the  methods  already 
considered,  we  get  j,  =  _  2,  /  =  _  4,  c  =  -  41. 

Hence  the  equation  of  the  circle  sought  is 

The  student  should  verify  directly  that  this  equation  is  satisfied  by  the 
coordinates  of  the  given  points. 

PROBLEMS 

1.  On  the  Centigrade  thermometer  the  temperature  of  melt- 
ing ice  is  marked  0°  and  that  of  boiling  water  100°.  Write 
down  the  equation  giving  the  relation  between  the  reading  of 
the  Centigrade  and  the  Fahrenheit  thermometers,  and  give  a 
graphical  means  of  changing  from  one  of  these  readings  to  the 
other  by  drawing  the  locus  of  this  equation. 

2.  The  following  temperatures  (Fahrenheit)  were  reported 
by  the  New  York  Swi  for  Jan.  16,  1912. 

Degrees                                                               Degbeks 
8  A.M 8 

9a.m 2 

10  A.M 6 

11  A.M 10 

Noon 12 

Ip.M 11 

2  p.M 13 

3p,m 13 

4  p.M 13 

From  the  figure  of  the  preceding  problem  measure  off  the 
corresponding  temperatures  on  the  Centigrade  thermometer. 

3.  A  train  passes  a  station  at  the  rate  of  10  miles  an  hour. 
Half  an  hour  afterwards  another  train  follows  it  at  the  rate  of 
12  miles  an  hour.  Determine  graphically  how  long  it  will  take 
the  latter  to  overtake  the  former. 

Let  y  =  number  of  miles  traveled  by  the  first  train  in  x  hours.  Then 
y  =  10  X.     The  locus  of  this  equation  is  the  line  marked  (1)  in  the  figure. 


1  A.M 

....     10 

2  A.M 

....      8 

3  A.M 

.     ...       8 

4  A.M 

....      6 

5  A.M 

....      4 

6  A.M 

....       4 

7  A.M 

.     ...      3 

7.30  A.M 

.     .     .     .       2 

EQUATIONS  IN  TWO  OR  MORE   UNKNOWNS       69 


It  represents  graphically  the  rela- 
tion between  the  distance  trav- 
eled and  the  time  of  traveling 
of  the  first  train.  When  the  first 
train  has  traveled  x  hours,  the 
second  one  has  traveled  x  —  ^ 
hours.  If  now  we  let  y  =  no.  of 
miles  traveled  by  the  second 
train,  y  =  12(x  —  I).  The  locus 
of  this  equation  is  the  line 
marked  (2)  in  the  figure.  The 
abscissa  of  the  point  of  intersec- 
tion of  these  two  loci  represents 
the  number  of  hours  from  the 
time  of  departure  of  the  first  train  to  the  time  of  meeting. 

4.  Determine  graphically  how  long  it  will  take  an  automo- 
bile going  25  miles  an  hour  to  overtake  another  one  that  had 
15  miles  start  and  is  going  at  the  rate  of  16  miles  an  hour. 

5.  Determine  graphically  where  two  trains  will  meet  if  one 
of  them  starts  from  Detroit  for  Chicago  at  the  rate  of  30  miles 
an  hour  at  the  same  time  the  other  one  starts  from  Chicago 
for  Detroit  at  the  rate  of  40  miles  an  hour.  The  distance  from 
Detroit  to  Chicago  is  285  miles. 

6.  A  man  rows  9  miles  downstream  in  2  hours  and  returns 
in  4  hours.  How  fast  does  he  row  in  still  water,  and  what  is 
the  rate  of  the  current  ? 

In  solving  this  problem  assume  that  the  rate  downstream  is  equal  to 
the  rate  of  rowing  in  still  water  increased  by  the  rate  of  the  stream,  and 
that  the  rate  upstream  is  equal  to  the  difference  of  these  two  rates. 

7.  A  and  B  working  together  do  one  fifth  of  a  piece  of  work 
in  a  day.  After  they  have  worked  together  three  days,  B 
stops  and  A  finishes  in  three  days  more.  How  long  would  it 
have  taken  each  one  to  do  the  work  alone  ? 

8.  A  glass  of  water  weighs  14  ounces.  When  full  of 
sulphuric   acid   of    specific    gravity  1.35,   the    glass  weighs 


70  COLLEGE  ALGEBRA 

17.5  ounces.     Eind  the  weight  of  the  glass  when  empty  and 
the  number  of  ounces  of  water  it  holds. 

See  the  Appendix  for  an  explanation  of  specific  gravity. 

9.  A  certain  vessel  weighs  p  pounds  when  full  of  a  sub- 
stance of  specific  gravity  Si,  and  it  weighs  q  pounds  when  full 
of  a  substance  of  specific  gravity  So.  What  is  the  weight  of 
the  vessel,  and  how  much  of  each  substance  does  it  hold  ? 

10.  Two  weights  of  25  pounds  and  30  pounds  respectively 
balance  when  resting  on  a  lever  at  unknown  distances  from  the 
fulcrum.  If  5  pounds  are  added  to  the  first  weight,  the  other 
one  must  be  moved  2  feet  further  from  the  fulcrum  to  main- 
tain the  balance.  What  was  the  original  distance  from  the 
fulcrum  to  each  of  the  weights  ? 

11.  Two  weights  of  w^  and  Wg  pounds  respectively  balance 
when  resting  on  a  lever  at  unknown  distances  from  the  ful- 
crum. If  p  pounds  are  added  to  the  first  weight,  the  other  one 
must  be  moved  /  feet  further  from  the  fulcrum  to  maintain  the 
balance.  What  was  the  original  distance  from  the  fulcrum  to 
each  of  the  weights  ? 

12.  A  weight  of  20  pounds  is  placed  at  random  on  a  beam 
of  unknown  length  that  is  supported  at  its  ends.  It  is  found 
that  this  produces  a  pressure  of  15  pounds  on  the  support  at 
one  end  in  addition  to  that  caused  by  the  weight  of  the  beam. 
It  is  also  found  that  the  same  pressure  is  produced  on  this 
support  by  a  weight  of  25  pounds  placed  2  feet  further  from 
this  end.     How  long  is  the  beam  ? 

13.  A  weight  of  p  pounds  is  placed  at  random  on  a  beam  of 
unknown  length  that  is  supported  at  its  ends.  It  is  found 
that  this  produces  a  pressure  of  a  pounds  on  the  support  at 
one  end  in  addition  to  that  caused  by  the  weight  of  the  beam. 
It  is  also  found  that  the  same  pressure  is  produced  on  this 
support  by  a  weight  oi  p-\-n  pounds  placed  m  feet  further 
from  this  end.     How  long  is  the  beam  ? 


EQUATIONS  IN  TWO  OR  MORE  UNKNOWNS       71 


14.  Represent  graphically  the  change  in  pressure  borne  by 
each  support  of  the  bridge  described  in  Problem  16,  p.  45,  as 
the  wagon  moves  across  the 
bridge. 

Let  X  =  the  number  of  tons  pres- 
sure due  to  the  wagon  on  the  first 
support. 

Let  y  =  the  number  of  feet  from 
the  first  support  to  the  wagon. 

Then  26 x  =2(25 -y), 

or  25x+2y=50. 

The  point  on  the  locus  of  this  equa- 
tion whose  ordinate  is  the  distance 

of  the  wagon  at  any  moment  from  the  first  support  has  for  abscissa  the 
pressure  in  tons  on  this  support  at  this  moment. 


15.  A  beam  18  feet  long  is  supported  at  its  ends.  Repre- 
sent graphically  the  change  in  the  pressure  on  each  support  as 
a  weight  of  150  pounds  moves  across  the  beam. 

16.  A  locomotive  weighing  600  tons  moves  across  a  bridge 
70  feet  long  that  is  supported  by  stone  abutments  at  the  ends. 
Represent  graphically  the  change  in  the  pressure  on  the  two 
supports. 

Find  the  equation  of  the  circle  that  passes  through  each  of 
the  following  sets  of  three  points.     Draw  each  circle. 


17.  (3,2),  (1,4),  (-2,1). 

18.  (0,  0),  (3,  0),  (0,  5). 


20.  (-5, -5),  (3,  3),  (4,0). 

21.  (-3, -6),  (2,7),  (4,8). 


19.   (-2,1),  (3,4),  (4, -2). 

22.  Find  the  distances  from  the 
vertices  to  the  points  of  contact  of 
the  circle  inscribed  in  the  triangle 
whose  sides  are  9,  14,  and  18 
respectively. 


72 


COLLEGE   ALGEBRA 


23.  Find  the  distances  from  the  vertices  to  the  points  of 
contact  of  the  circle  inscribed  in  the  triangle  whose  vertices 
are(-21, -22),  (11,2),  (-14,2). 

24.  Three  circles  are  drawn  with  centers  at  the  points 
(_26,  -39),  (46,  -9),  and  (-10,24)  respectively  and  such 
that  they  are  tangent  to  one  another  externally.  What  are  the 
equations  of  these  circles  ? 

25.  Find  the  equations  of  the  three  circles  that  are  tangent 
to  one  another  externally  and  have  their  centers  at  the  points 
(-16,  -6|),  (5,  2^),  and  (-5,  12). 

26.  In  the  triangle  ABC, 
AB  —  5  inches,  BC  =  8  inches, 
and  CA  =  10  inches.  What  must 
be  the  radii  of  three  circles  with 
centers  at  A,  B,  and  C  respec- 
tively in  order  that  those  whose 
centers  are  at  B,  C  shall  be  tan- 
gent to  each  other  externally  and 
shall  each  be  tangent  internally 
to  the  one  whose  center  is  at  ^  ? 

27.  In  the  triangle  ABC,  AB  =  c,  BC=a,  CA=b.  What 
must  be  the  radii  of  three  circles  with  centers  at  A,  B,  and  C 
respectively  in  order  that  those  whose  centers  are  at  B  £tnd  C 
shall  be  tangent  to  each  other  externally  and  shall  each  be 
tangent  internally  to  the  one  whose  center  is  at  -4  ? 


CHAPTER   VI 
FRACTIONAL  AND  NEGATIVE  EXPONENTS.     RADICALS 

54.  The  fundamental  laws  of  exponents.  —  The  funda- 
mental laws  in  the  use  of  positive  integral  exponents  are  ex- 
pressed by  the  formulae : 

I.  «"• .  a"  =  a*"+". 

II.  a"^  -i-a''  =  a'"~".     (When  n  is  less  than  m.) 

III.  {a^^Y  =  a"*". 

IV.  (ab)"'  =  a'^b"'. 

\bj       6- 

55.  Fractional  and  negative  exponents.  —  Now  heretofore 
we  have  considered  only  positive  integral  exponents,  since 
the  purpose  of  an  exponent  was  to  indicate  how  many  times 
a  number  was  to  be  taken  as  a  factor.  It  turns  out  however 
to  be  possible  to  use  fractional  and  negative  exponents  in  a 
simple  and  natural  way  that  is  also  consistent  with  these  laws. 
For  example,  if  we  agree  that  a^  shall  be  a  symbol  for  a  num- 
ber such  that  1      1 

a^  .  a^  =  a, 

we  shall  have  a  definition  that  is  consistent  with  the  first  of 
these  laws.  But  this  is  equivalent  to  saying  that  a^  is  one  of 
two  equal  factors  of  a.  We  accordingly  agree  that  this  symbol 
shall  represent  the  same  number  as  Va. 

More  generally,  if  p  and  q  are  any  two  positive  integers  and 

£ 

we  agree  that  a*  shall  be  a  symbol  for  a  number  such  that 
p     p 
a«  •  a*  •••  (to  5  factors)  =  a^, 

73 


74  COLLEGE  ALGEBRA 

we  shall  have  a  definition  that  is  consistent  with  the  first  law. 

p 
But  this  makes  a^  one  of  q  equal  factors  of  a^,  or  a  gth  root 

p 
of  aF.     We  agree  then  that  a'  shall  be  a  symbol  for  a  number 

such  that  p        _ 

It  would  be  in  accordance  with  the  fundamental  laws  of  exponents  to 
agree  that  92  =—  3.  But  in  view  of  the  fact  that  a  positive  number  has 
only  one  positive  even  root  and  that  any  real  number  has  only  one  real 
odd  root  it  is  agreed  to  call  the  positive  even  root  of  a  positive  number 
and  the  real  odd  root  of  a  negative  number  principal  roots.  Radical  signs 
and  fractional  exponents  are  used  to  indicate  these  principal  roots.  Thus, 
we  say  that  9^=  V9  =  3,  and  -  9^  =  -  V9  =  -  3. 

It  should  be  noted  that  these  remarks  do  not  apply  to  even  roots  of 
negative  numbers.     (See  §  133) . 

56.   Suppose  now  that      x  =  -Va, 
and  y  =  ^b. 

Then  af  =  a, 

and  ]f  =  b. 

Hence,  x^y^  =  ab, 

and  (xyy  =  ab.  (Law  IV.) 

Therefore,  xy  =  \Jab^ 

or  ■V~a'^b  =  ^'ab, 

11  1 

or  a^'¥  =  {ciby. 

A  direct  application  of  this  relation  shows  that 

1  1  _  :? 

{cfiy  -  (iapy  =  VcP'  =  a*. 

p  1 

Hence  a'  may  be  looked  upon  as  the  pth  power  of  a'  as  well 

as  the  qth.  root  of  a^. 
For  example, 
9^  =  V9  =  3,  2^  =  v/23  =  y/S,  (a  -  6)^  =  \/(a-by  =  (  ^a^^)*. 

Note.  — The  symbol  ai  is  read  "  a  to  the  power  p  divided  by  g,"  or 
*'  a  exponent  p  divided  by  g." 


FRACTIONAL  AND  NEGATIVE  EXPONENTS        75 

If  a  and  b  are  negative  and  q  is  even,  care  must  be  taken 
1  1 
in  forming  the  product  a*  •  ¥.  For  example,  we  would  natu- 
rally say  that  (-  1)^  •  (-  1)^  =  (-  1  •  -  1)^  =  (1)^  =  1.  But, 
as  we  shall  see  in  §  133,  we  interpret  the  symbol  (—  1)^  in 
such  a  way  that  (— 1)^  •  (— 1)^  = -1. 

57.   If  we  agree  that  a^  shall  be  a  symbol  for  a  number  such 
that 


that  is,  if  we  agree  that  this  symbol  shall  obey  the  first  law 
of  exponents,  we  must  say  that 


0      a        i 
a"  =  —  =  1. 


It  is  agreed  that  this  relation  shall  hold  for  all  values  of  a 
except  0.     We  do  not  give  any  meaning  to  the  symbol  0^. 

Also,  in   accordance  with   the   first   law  of  exponents,  we 

agree  that  a~'"{m  a  positive  number)  shall  be  a  symbol  for  a 

number  such  that 

a"*"  .  «"»  =  a^  ==  1. 

This  is  equivalent  to  saying  that  a  number  with  a  negative 
exponent  or,  as  we  shall  say,  raised  to  a  negative  power,  shall 
be  equal  to  the  reciprocal  of  this  number  raised  to  the  corre- 
sponding positive  power, 

«-"'  =  —  . 
a"* 

With  this  agreement,  the  value  of  a  fraction  will  not  be 
affected  if  any  factor  is  transferred  from  the  numerator  to  the 
denominator,  or  from  the  denominator  to  the  numerator,  pro- 
vided the  sign  of  its  exponent  is  changed. 

Thus, 

a.2  ^a;2?/-3^      1 .   {x  +  2  yY(x  -  3  y)^  ^  (x  +  2  yY{x  -3  y)5(a-6)-3 

^3^4        z^        x-^yH^ '     (a  -  6)3(a  +  4  6)2  (a  +  4  6)2 

=  (x  +  2  yy^x  -  3  yy(a  -  b)-^{a  +  4  6)-2. 

Vi 


76  COLLEGE  ALGEBRA 

The  student  is  cautioned  that  this  applies  only  to  factors  of 
the  numerator  and  of  the  denominator,  and  not  to  separate 
terms. 

Thus,  ?!-+l!  is  not  equal  to  -^  or  to       ^* 


3^'^  y-a  +  z^ 

58.  We  have  agreed  upon  a  use  of  fractional,  zero,  and 
negative  exponents  that  is  consistent  with  the  first  law  of  ex- 
ponents. It  can  be  shown  that  this  use  of  these  exponents  is 
consistent  with  all  five  of  the  fundamental  laws  of  exponents. 
It  is  even  possible  to  use  irrational  exponents  in  a  way  con- 
sistent with  these  laws.  Hence  we  can  consider  these  laws, 
as  given  in  §  54,  as  applicable  to  all  real  exponents. 

This  use  of  fractional,  zero,  and  negative  exponents  merely 
gives  us  new  ways  for  indicating  operations  with  which  we 
were  already  familiar. 

If  we  were  to  use  these  exponents  in  a  way  not  consistent 
with  the  fundamental  laws  of  exponents,  the  usefulness  of 
algebra  as  a  means  for  getting  information  would  be  seriously 
impaired. 

EXERCISES 

Write  each  of  the  following  expressions  without  the  use  of 
the  radical  sign : 

1.    Vl9.  4.    ^■^J{x^ryf. 


2.    v'24.  5.    \V^fz-\ 


3.    V3l  6.    ^{x-Zyf{^x^^y). 

8.    i#. 
36\62 


9      4/(4  a  + 3  6)^(2  g-r^y^ 
•    \  {x-Vyfi?x-2yf 

10.  vn. 


FRACTIONAL  AND  NEGATIVE  EXPONENTS        77 

Write  each  of  the  following  expressions  without  the  use  of 
fractional  or  negative  exponents  : 

11     A.  14.    8~i 

■  ^*  X5.  (-*)-f. 

12.  x^y-i  16.    S^V^-l 

13.  3^xi  17.    G\)i 

18.  (o^-{-2x'y-5xy^-{-3f)i 

19.  r i._V^ 

.   •  20.    (2a;-3  2/)^(3a;-2  2/)^(2a;  +  3  2/)'^. 

Write  each  of  the  following  expressions  without  negative 
exponents  and  in  as  simple  a  form  as  possible : 


21. 

(ff)^. 

22. 

(.64)-l 

23. 

it)-'- 

24. 

8456". 

25. 

a  +  a-i 

a  —  tt~^ 

26. 

2  —  2-2 

2  +  2-2 

27. 

1 

X-'  +  2/"' 

Multiply : 

28. 

7x-*. 

29. 

(7x)-\ 

30. 

(64  a-'b'c^yi 

31. 

2 

32. 

33. 

1  ,  (e^-e-y 

^  '          4 

34.  a^  +  a^6^  +  5  by  ci^  _  ^i. 

35.  Va^  +  Vi/*  by  a;^  —  2/^. 

36.  m^  +  m^n^  +  n^  by  m^  —  n^. 


78  COLLEGE  ALGEBRA 

Divide : 

37.  a^-\-b^hj  a^  +  bK 

38.  16  m2  -  81  n^  by  2  m^  -  3  riK 

39.  15a^-ah^-QbhyS-^a-2Vb. 
Expand : 

40.    (a^  —  x^y,  41.    (a^  —  x^^y. 

59.  Radicals.  —  A  radical  is  any  algebraic  expression  of  the 
form  -y/a,  or  b-y/a,  where  n  is  a  positive  integer.  In  such 
expressions  a  is  called  the  radicand,  ri  the  index,  and  b  the 
coefficient. 

A  radical  is  said  to  be  in  its  simplest  form  when : 

(a)  The  radicand  is  an  integer,  or  a  polynomial  with  integral 
coefficients  if  it  contains  any  literal  terms. 

(b)  The  radicand  contains  no  factors  of  the  type  indicated 
in  (a)  raised  to  powers  equal  to,  or  greater  than,  the  index  of 
the  radical.  ^^  ^  -"■ 

(c)  The  radicand  is  not  a  power  whose  index  has  a  factor  in 
common  with  the  index  of  the  radical. 

Thus  the  radicals  V|,  V(a  —  a)*,  and  Va^  are  not  in  their  simplest 
form. 

60.  Simplification  of  radicals.  —  Certain  radicals  can  be  sim- 
plified by  means  of  one  or  more  of  the  following  reductions. 
But  not  every  radical  can  be  made  to  assume  the  simple  form 
described  in  the  preceding  article. 

(1)  Reduction  of  a  fractional  radicand  to  the  integral  form. 

This  reduction  can  always  be  performed  as  follows :  if  the 
radicand  is  not  a  single  fraction  in  its  lowest  terms,  put  it 
into  this  form.  Then,  if  the  radical  is  of  index  p,  make  the 
denominator  of  the  radicand  a  perfect  pth  power  by  multiply- 
ing the  numerator  and  the  denominator  of  the  fraction  by  a 
properly  chosen  expression.     The  original  radical  is  equal  to 


FRACTIONAL  AND  NEGATIVE   EXPONENTS        79 

the  pth  root  of  the  resulting  numerator  divided  by  the  pth. 
root  of  the  resulting  denominator,  which  is  rational. 


Thus,  J^  =  Jl  =  ^, 

'  \3      ^9        3 


and  in 


Pla      P  labP-^      K/abP-^ 


general,     ^'^  ^  ^^^^^  =  voo^^    (Fundajnental  Law  V.) 
^b       ^    bp  b 


(2)  The  removal  of  factors  from  the  radicand. 

This  reduction  can  be  made  only  when  the  radicand  con- 
tains factors  to  powers  equal  to,  or  greater  than,  the  index  of 
the  radical.  The  following  examples  illustrate  the  method 
of  procedure  and  are  based  upon  Fundamental  Law  IV. 

Examples.       VS  =  \/4T2  =  Vi  •  V2  =  2  V2. 

</m  =  </W^=y/27-  ^  =  3v^. 

y/ia-by  =  \/(a  -  6)«  •  y/a^^  ={a-b) V{a  -  6). 

(3)  The  lowering  of  the  index  of  the  radical. 

When  the  radicand  is  a  power  whose  index  has  a  factor  in 
common  with  the  index  of  the  radical,  the  radical  is  equal  to 
another  radical  of  lower  index. 

Example.       Va*  —  d^  =  a^  =:  yfafi. 

In  certain  problems  it  is  desirable  to  introduce  factors  into 
the  radicand  or  to  increase  the  index  of  the  radical.  These 
reductions  are  the  inverses  of  (2)  and  (3)  respectively. 

Examples.  4  ^  =  W^  •  v^  =  y/US. 

61.   Addition  and  subtraction  of  radicals.     Definition.  —  Two 

or  more  radicals  are  said  to  be  similar  if,  when  simplified,  they 
have  the  same  index  and  the  same  radicand. 

For  example,  2\^  and  3V5  are  similar,  as  are  also  V^b  and  SVab. 


80  COLLEGE  ALGEBRA 

An  expression  involving  two  or  more  radicals  can  sometimes 
be  simplified  by  first  simplifying  each  radical  and  then  com- 
bining the  similar  radicals  in  the  way  illustrated  in  the  follow- 
ing examples : 

Examples.       2\/98  -  3  V50  +  V72  =  14  V2  -  15  V2  +  6  V2  =  5  V2. 

2 V98-60 VS-H  V32-  Vl08  =  14\/2-  50V3  +  4 V2  -  6 V3 
=  18V2-56\/3. 

It  should  be  observed  that  the  sum  or  difference  of  two  dis- 
similar radicals  cannot  be  expressed  as  a  single  radical. 

62.  Multiplication  of  radicals.  —  The  product  of  two  radicals 
with  a  common  index  is  a  radical  with  the  same  index  whose 
coefficient  and  radicand  are  equal  respectively  to  the  products 
of  the  coefficients  and  of  the  radicands  of  the  factors.  (Funda- 
mental Law  IV ;  see  §  56.) 

Thus,  ay/b  •  cVd  =  ac  y/bd. 

If  the  radicals  do  not  have  the  same  index,  they  should  first 
be  reduced  to  equal  radicals  with  a  common  index. 

Thus,  \/5  .  V6  =  v^  .  V2I6  =  v^5400. 

The  product  of  such  expressions  as  Va  +  V&  — Vc  and 
•\/a  —  ^/b-\-Vc  can  be  found  by  applying  the  usual  rules  for 
the  multiplication  of  polynomials  in  connection  with  the 
principles  just  stated. 

Thus,  Va  +  Vb  -Vc 

Va  —  Vb  -f- Vc 
a  +  y/ab  —  Vac 
—  Vab  —  b  -^-Vbc 

■f  Vac         +  Vbc  —  c 


a  —  b  -{■  2  Vbc  —  c 


It  is,  in  general,  best  to  simplify  all  the  radicals  in  a  problem 
before  attempting  to  perform  any  operations  with  them. 


FRACTIONAL  AND  NEGATIVE  EXPONENTS        81 

63.  Division  of  radicals.  —  The  quotient  of  two  radicals  with 
a  common  index  is  a  radical  with  this  same  index  whose  co- 
efficient and  radicand  are  equal  respectively  to  the  quotients 
of  the  coefficients  and  of  the  radicands  of  the  dividend  and 
the  divisor.     (Fundamental  Law  V.) 


Thus,  aVb^cVd  =  ^^ 

c  ^d 


If  the  radicals  do  not  have  the  same  index,  they  should  first 
be  reduced  to  equal  radicals  with  a  common  index. 

64.  Rationalizing  the  denominator.  —  An  algebraic  expres- 
sion in  the  form  of  the  indicated  quotient  of  two  polynomials 
is  called  a  rational  function  of  the  letters  contained  in  it.  If 
the  polynomial  in  the  denominator  does  not  contain  any  of 
the  letters,  the  expression  is  called  a  rational  integral  function 
of  the  letters. 

Thus,    — -^ ^—^ — ^    is  a  rational   function    of  x   and   y,   while 

4:X  —  6y 

7  ic2  _  6 ajy  +  2  ?/2  and  ^  "*"_  ^    are  rational  integral  functions. 

V2 

When  we  say  that  an  expression  containing  no  letters  is 
rational,  we  mean  that  it  is  a  rational  number;  that  is,  the 
quotient  of  two  integers. 

Certain  fractions  whose  denominators  are  irrational  can  be 
changed  into  equal  fractions  with  rational  denominators. 

For  example, 

3    _     3V2        3V2      (See  §60,1.) 


V2      V2  .  V2         2 


■y/a  —  Vb  _  a(Va  —  Vb) 


Va  +  Vb      Va-\-Vb     ^a-y/b  «-^ 

Such  a  change  can  always  be  made  when  the  denominator 
of  the  fraction  is  a  single  radical  or  the  sum  of  two  terms,  one 
of  which  is  a  square  root  and  the  other  either  a  square  root  or 


82  COLLEGE  ALGEBRA 

rational.  Of  course  such  a  change  is  possible  in  many  other 
cases  also. 

If  the  product  of  two  irrational  expressions  is  rational,  either 
of  the  expressions  is  called  a  rationalizing  factor  of  the  other. 

In  computing  the  numerical  value  of  fractions  with  irrational 
denominators  it  is  best  first  to  rationalize  the  denominators 
whenever  this  is  possible. 

EXERCISES 

Simplify  the  following : 

1.    Vi8.  8.   Va^T3a¥+3^FTP. 


■4 


2.  </l. 

9 

3.  V45. 

5.  3V(a;-2/)^  ^h 

6.  -VSia-^-hf.  11.   3Vi-V50  +  4Vl62. 

7.  a^ftl  12.   3a;^-2a:l 

13.  3V80-2V75  +  V108  +  5V20. 

14.  3Va«-2Vi^2  +  4V^l 

15.  3V-^-5VS-2V|. 

16.  ■y/Ja^-h') {o? J^2ah  +  W)  +  -y/^^~^. 

17.  Which  is  the  greater,  -^9  or  V5  ? 

Hence  V5  >  ^9. 

18.  Which  is  the  greater,  VlO  or  -^28  ?   V3  or  -v^6  ?   Vl9 
or  -y/eE? 

Perform  the  following  indicated  multiplications : 

19.  3V45.4V72.  21.    -^^  •  ^f 

20.  6ax^'5ax~K  22.    V(a  -  6)^  Va  +  6)1 


FRACTIONAL  AND  NEGATIVE  EXPONENTS   83 

23.  (Va-Va^)^    24.  (10  -  V5)  (5  +  VlO). 

25.  (  V2  +  V3  +  V5)  (  V  2  -  V3  -  V5). 

26.  {</2-\-y/7t){-</I--</2^ -{-</' a'). 

27.  Is  1  —  V3  a  root  of  the  equation  ocr  —  x  — 1=0? 

28.  Is  -^-^^^  a  root  of  the  equation  x^-]-3x-i-l  =  0? 

Simplify  each  of  the  following  expressions: 

29.  V35--Vf  VrT^-a^(l  +  a.T^ 

30.  6-f-V2.  36.  1  +  a^^ 

31.  5-^/4.  J\. 


X Y 


32.  2Va^V2VL  ^^1  + 

33.  -^i?6?-^^^.  ^^  0^(1 -a^-t 

34.  (V8+5VT2-3Vi4)-2.  *         1  //      1      y"^ 

35.  (V2+V3- V5)-V6.         vT^^^wrr^v 


Vl+a.-2-l 


38.  -  ^^  +  ^ 


39. 


^a  —  x  —  ^a  -{-x 


Reduce  each  of  the  following  fractions  to  an  equal  fraction 
with  a  rational  denominator : 

40. 


4 

11     V5-V2 

lo    1+V6 

V5-2 

V5+ V2 

1- V6 

43.         _  ._  _ 

Va  +  V6  Va;  — 1 


p  •  1 5 


s 


CHAPTER   VII 
QUADRATICS 

65.  Quadratic  equations.  —  Sometimes  our  information  con- 
cerning a  number  comes  to  us  in  the  form  of  an  equation  of 
the  second  degree  in  the  unknown.  Such  an  equation  is  known 
as  a  quadratic  equation.  In  this  chapter  we  shall  consider  how 
to  identify  a  number  described  by  such  an  equation. 

Consider  the  equation       6x^  —  x  —  2  =  0.  (1) 

We  observe  that  the  left  member  can  be  factored  and  that  the  equation 
can  be  written  in  the  form 

(3a;-2)(2a;  +  l)  =  0.  (2) 

Now  we  know  from  Theorem  10,  Chapter  I,  that  if  x  is  any  number 
that  satisfies  this  equation,  it  must  be  such  that  either 

3  X  -  2  =  0,  (3) 

or  2x4-1=0;  (4) 

and  conversely,  any  number  of  this  kind  must  satisfy  the  given  equation. 
That  is,  the  numbers  described  by  (1)  are  the  same  as  the  numbers  de- 
scribed by  (3)  and  (4). 

In  general,  if  the  right  member  of  an  equation  in  one  un- 
known is  zero  and  the  left  member  is  factored,  any  number 
which  when  put  in  place  of  the  unknown  makes  one  of  these 
factors  zero  must  satisfy  the  equation,  and,  conversely,  any  root 
of  the  equation  when  put  in  place  of  the  unknown  must  make 
one  of  these  factors  z6ro.     Hence, 

66.  In  order  to  solve  an  equation  in  one  unknown,  transpose 
all  the  terms  to  the  left  member  of  the  equation,  making  the  right 
member  zero;  factor  the  resulting  left  member;  put  equal  to  zero 

84 


QUADRATICS  85 

each  of  these  factors  that  contains  the  unknown  and  solve  the  re- 
sulting equations,  which  will  be  of  lower  degree  than  the  original 
equation. 

This  method  of  solving  an  equation  is  an  important  one  and 
is  applicable  to  equations  of  any  degree.  In  practice,  how- 
ever, it  is  subject  to  limitations  because  there  are  so  many 
polynomials  in  one  variable  whose  factors  we  cannot  find. 

67.  Equations  with  given  roots.  —  With  Theorem  10  of 
Chapter  I  in  mind  it  is  easy  to  form  an  equation  that  shall 
have  given  numbers  for  roots. 

If,  for  example,  the  given  numbers  are  ri,  r2,  •••  Vn,  it  is  obvious  that 

(X  -  ri)(x  -  Vi)   •••   (X-  Tn)  =  0 

is  an  equation  whose  roots  are  these  numbers. 

EXERCISES 
Solve  the  following  equations  by  factoring,  and  check  your 
results : 

1.  ar^- 1=:0.  9.  y^-20  =  y. 

2.  ic2_2a;+i==o.  10.  2x'-llx  +  12  =  0. 

3.  a^-2x^-\-x  =  0.  11.  522  +  29:^-6  =  0. 

4.  x--\-9x-\-S  =  0.  12.  0  =  a;  +  4a^. 

5.  X*- 13  0^  +  36  =  0.  13.  0  =  m2  +  5m4-6. 

OS.   x^  +  6x^  +  llx  +  6  =  0.         14.   x^-^3a^-33x-35  =  0. 

7.  ix(x-5){x-\-7)=0.  15.   x^-^6x'-\-2x=30-\-3x. 

8.  y^  =  25.  16.    a^  +  2ox  +  144  =  0. 

Form  an  equation  whose  roots  are : 

17.   2,3.  21.    I,  l,f. 


18.    -1,5.  oo     -1+V5    -1-V5 

2        '  2 


22. 
23. 


-3  +  V41    -3-V41 


86  COLLEGE  ALGEBRA 

68.  Solution  by  completing  the  square.  —  When  we  have 
difficulty  in  applying  this  method  to  the  solution  of  a  quadratic 
equation  we  can  proceed  as  illustrated  in  the  following  examples : 

(a)  Solve  the  equation 

3x2  + 2  a;  =  4.  ^^1^ 

Divide  each  member  by  3, 

x^  +  ^x  =  i.  (2) 

Add  (i)"-^  to  each  member, 

a;-  +  I  ^  +  i  =  ¥•  (3) 

Extract  the  square  root  of  each  member, 


Check.  3^-l±Vl3y^, 


-  1  ±  \/l3 
3 


1  T2v/13+  13      -2 +  2v/l3 
3  3 


(6)  Solve  the  equation 

Transpose  4  to  the  left  member, 

Sx^-\-2x=-4. 

Divide  each  member  by  3, 


Add  (^)2  to  each  member, 

Extract  the  square  root  of  each  member, 

^  I  1  -  ,  v^^ 

"^+3"^       3      ' 


QUADRATICS  87 

Check.  .3(=l±^^HnV+2.3±±^E3!  +  4 


(=^±i^) 


^  1  =p2V-ll-ll      -2±2V-  11  _^  12 
3  3  '3 

=  0. 

In  simplifying  l~ — ^        —  j    the  student  should  bear  in  mind  that 

y/ZTTl .  V^ni  =  -  11.     (See  §  133.) 

The  terms  involving  the  unknown  x  should  be  brought  to 
the  left  side  and  the  constant  terms  to  the  right  side  of  the 
equation.  If  the  coefficient  of  x^  is  not  a  perfect  square,  it 
should  be  made  such  by  multiplying  or  dividing  each  member 
of  the  equation  by  a  properly  chosen  constant.  The  term  in 
X  should  then  be  divided  by  twice  the  square  root  of  the  term 
in  x^,  and  the  square  of  the  quotient  added  to  each  member 
of  the  equation.  The  resulting  left  member  will  be  a  trino- 
mial square.     Why  ? 

The  roots  of  the  equation  can  now  be  found  by  putting  one 
of  the  square  roots  of  the  left  member  equal  in  turn  to  each 
of  the  square  roots  of  the  right  member  and  solving  each  of 
the  resulting  linear  equations. 

This  method  of  solving  a  quadratic  equation  is  known  as 
the  method  of  solution  by  completing  the  square. 

69.  Equation  (3)  of  the  first  illustrative  example  can  be 
written  thus  ^.  +  ^  ^+ ,  _  ,^  =  0. 

The  left  member  of  this  equation  can  be  looked  upon  as  the 
difference  of  two  squares  and  factored  accordingly. 


hl-mhl-m-^- 


Equating  each  of  these  factors  to  zero,  and  solving  the  resulting 
equations,  we  get 


x  =  ^i^-^andx=^i±^3 


as  before. 


88  COLLEGE  ALGEBRA 

Thus  the  operation  of  completing  the  square  in  a  quadratic 
equation  may  be  considered  merely  as  a  device  for  enabling 
us  to  see  the  factors  of  the  left  member  of  the  equation  when 
the  right  member  is  zero.  And  therefore  the  method  of  solu- 
tion by  completing  the  square  is  essentially  a  part  of  the 
method  of  solution  by  factoring.  It  should  be  observed,  how- 
ever, that  factors  obtained  in  this  way  are  not  always  of  the 
kind  considered  in  Chapter  II.     (See  §  18.) 

EXERCISES 
Solve  each  of   the  following  equations   by  completing  the 
square,  and  check  your  results : 


1. 

Sx'=2x  +  5. 

17. 

5a;2  +  13  =  0. 

2. 

2x'-\-5x-3  =  0. 

18. 

x'  =  7x. 

3. 

5a^  +  3a;  =  3. 

19. 

32/^-2  =  0. 

4. 

7y  +  2=:5f. 

20. 

3  ,^  _  ^  =  0. 

5. 
6. 

4.f  =  t-h3. 

a^     (Sx  +  by_^ 
4^9 

4r2  +  6r4-l  =  0. 
2y'-^y-3  =  0, 
a^  =  3-x. 

21. 
22. 

x^-^x-l  =  0. 
x'-x-l=0. 

7. 
8. 
9. 

23. 
24. 

9^4 
a^  +  l  =  0. 

10. 

(mo; + 3)2  =  4  X. 

25. 

(3x-\-iy  =  Sx. 

11. 

aj2-f  a;  +  l  =  0. 

26. 

52/^  +  52/4-1=0. 

12. 

a^-x  +  l  =  0. 

27. 

(x-^lf-Q^==l. 

13.  (2x-3y-h{x-iy=3x+4:.  28.  x'  +  ax  =  2, 

14.  lx^-\-mx-\-n  =  0.  29.  (k -]- l)y^ -^  Jcy -\- 3  =  0. 

15.  6.2-13. +  6  =  0.  30^     2(0^-2)^^-^ 

16.  ^  =  4..  '2  3 


QUADRATICS  89 

70.   Solution  by  formula.  —  The  equation 

is  called  the  general  quadratic  equation,  since  it  reduces  to  any- 
given  quadratic  when  proper  values  are  given  to  a,  6,  and  c.  The 
solution  of  this  general  quadratic  will  therefore  give  us  a  formula 
from  which  we  can  write  down  the  roots  of  "any  quadratic. 

Now  the  student  can  verify  by  using  the  method  of  com- 
pleting the  square  that  

—  h± ^W  —  4  flc 


2a 

are  the  roots  of  this  equation. 

Before  this  formula  is  applied  to  the  solution  of  a  given 
equation,  the  equation  must  be  simplified  and  all  its  terms 
brought  into  the  left  member. 

Suppose  for  example  that  we  want  to  solve  the  equation 
2  1^2 

X     X  -\-  2      5 
Clear  of  fractions,        10  x  -{-  20  +  6x  =  2  x^  +  4  x. 
Transpose  all  the  terms  to  the  left  member, 
-2ic2  + llx  +  20  =  0. 
Then  by  the  formula  (a  =  —  2,  ft  =  11,  c  =  20) 


Check. 


-  11±V121+  160 

-11 

±  V  281 

-4 

-4 

-  11  ±  V281      - 
-4 

1 
11±V281  1   « 

-     -»     +- 

-  11  ±  V281      - 

-4 

-19±\ 

/281 

_    152T8V281 -f  44^4> 

/281 

(-11±V281)(- 

-19±^ 

/281) 

_196T12V281 

490  --f  30V281 

_2(98qF6>/281)  _ 
6(98=F6V281) 

.2 
5* 

90 


COLLEGE  ALGEBRA 


EXERCISES 
Solve  the  following  quadratics  by  means  of  the  formula: 


1.  x'-{-Sx-^l  =  0. 

2.  2y^-5y-3  =  0. 

x  +  1      x'  —  C) 


4. 


3  4 

5.  Ix^  4-  wx  +  n  =  0. 

6.  mx"^  —  3  a;  4-  c  =  0. 

7.  x'^  +  ax  +  b  =  0._ 


10. 


9.  42/2_5?/  =  0. 

10.  7n~  —  6  m  4-  5  =  0. 

n.  (8m-6)2  =  256m2. 

12.  A:a;2  _  3(A:  +  l)a^  +  5  =  0. 

13.  9(A;  + 1)2  =  20  A:. 

14.  (^-1)2  =  3(2  77-3). 

15.  64m2-256(l4-m2)=:0. 


8.    Sx-\-6x 


7  -\-x 

~2"  ' 


i  -\-  X 


16.    {mx-\-ky  =  6x. 


18    (x±5y      (S^-J^^^ 
9       ^        16 


+  3(p2^0. 


17.   x2  4- 


4. 


27. 


28. 


16 


19.  17y  +  4:  =  Sy\ 

20.  a;2-16  =  0. 

21.  (o2/  +  2)2  +  2/2  =  |. 

22.  bx'^  +  bx  =  l. 

23.  (aj-2)(7a;-l)  =  5. 
;£_-2)(mi»-l)=5. 

30.  x'^  —  Sx-  m{x -{-2x'^-i-^)  =  5x^-{-3. 

31.  Show  that 

aa^  -\-bx-{-c  =  a(  X 


25.  cp2_^irtV  +  8maj  +  64  =  9. 

26.  mV  +  8ma;+64  =  6a;.      - 


(^^±^  =  1. 

=  1.    -' 


16  9 

x^      (3X  +  5y 


9 


29.    6a;2_|_(^2^_'  =  i2. 


6  4_V62-4 


2a 


—  b  —  -Vb'-^  —  4  ac 
2  a 


71.  Nature  of  the  roots  of  a  quadratic.  —  Certain  things 
about  the  roots  of  a  quadratic  equation  can  be  determined 
without  solving  the  equation.     If  the  equation  is  in  the  form 

aa;2  4-  &x  +  c  =  0, 


QUADRATICS  91 

the  roots  are  —b±  -\/¥  —  4:ac 

2  a 
Hence,  when  a,  b,  and  c  are  rational  numbers, 

I.    lfb'^  —  4:  ac  =  0,  there  is  oyily  one  root ;  namely,  — 

In  this  case  it  is  customary  to  say  that  there  are  two  equal  roots. 

II.    Ifb^  —  4:  ac  is  the  square  of  a  rational  number  different 
from  zero,  the  roots  are  real,  unequal,  and  rational. 

III.  If  b^  —  4:  ac  is  positive  and  not  the  square  of  a  rational 
number,  the  roots  are  real,  unequal,  and  irrational. 

IV.  If  b"^  —  Aac  is  negative,  the  roots  are  imaginary  and  un- 
equal. 

The  expression  6^  —  4  ac  is  called  the  discriminant  of  the 
equation. 

Thus  the  roots  of  the  equation 

are  real,  distinct,  and  irrational,  since  its  discriminant  equals  13. 
If  we  call  the  two  roots  of  the  equation 

ax^  +  6.7;  H-  c  =  0 

a?!  and  x^  respectively,  so  that 


0^1=- 

-b-^-Vb'- 
2a 

4  «c       ,  ^  _-b- 

Wb^- 

4 

ac 

2a 

i 

x^  4-  x^  = 

=  -^^,  and 
a 

..  ^  _  ^"  -  (&'  -  4  ac)  _  c 
4a^                a 

nee,  the 

sum  of  the 

roots  of  the  equation 
ax'  +  bx-^c  =  0 

is  equal  to  minus  the  coefficient  of  x  divided  by  the  coefficient  of  x^, 
and  the  product  of  the  roots  is  equal  to  the  constant  term  divided 
by  the  coefficient  of  x^.     (See  §  150.) 

For  example,  the  sum  of  the  roots  of  the  equation 
is  —  I  and  the  product  is  ^. 


92  COLLEGE  ALGEBRA 

The  student  should  verify  this  statement  by  solving  the  equation  and 
forming  the  sum  and  the  product  of  the  roots. 

If  the  equation  is  not  in  the  standard  form,  it  must  be  reduced  to  this 
form  before  this  rule  is  applied. 

Consider,  for  example,  the  equation 

3  cc2  +  1  z=  5  a;  -  4  ic2. 
Transposing,  7  x^  —  6  x  -{-  1  =  0. 

Hence,  the  «um  of  the  roots  is  ^  and  the  product  is  ^. 

EXERCISES 

Determine  without  solving  the  equations  the  nature  of  the 
roots  of  each  of  the  following  equations.  Also  find  the  sum 
and  the  product  of  the  roots  of  each  equation : 

1.   2aj2H-7x4-l  =  0.  8.   x'  =  10x-25. 

9  10.  x'  +  6x-\-9  =  0. 

3.  x''-x-\-2  =  0.  11.  (^x  +  iy  =  5(x-\-2). 

4.  x''-x-2  =  0.  12.  {2  +  xy  =  {l  +  2xy. 

5.  (Sx-\-5y  =  6x.  13.  4m2-6m  +  5  =  0. 

1*4  9  *  15.    m2  +  7  =  0. 

[    7.    5x^-\-ox  =  l.  16.   x(—2x-^S)  =  5. 

17.  x'-\-2x{2x-l)  +  {2x-iy-{-x-^{2x-l)  +  l  =  0, 

18.  {-x-^2y  +  6(-x-{-2)  +  9  =  7{x-S). 

19.  aj2  +  3a;  +  9  =  0.  20.    a;^- 3  a; +  9  =  0. 

For  what  values  of  k  will  the  roots  of  each  of  the  following 
equations  be  equal  ? 

21.   x^-4:kx-\-4:  =  0. 

Here  o  =  1,  6  =  -  4  A:,  and  c  =  4.     Hence,  b^  -  4tac  =  IQ  k'^  -  16. 
In  order  that  the  roots  be  equal,  we  must  have 

16A;2-16  =  0. 


QUADRATICS  93 

Solving  this  equation  for  k  we  get  A;  =  ±  1.  That  is,  when  k  in  this 
equation  has  the  vahie  1 ,  or  —  1 ,  the  equation  has  equal  roots.  We  can 
readily  verify  this  result.     When  A:  =  1,  the  equation  is 

a:-4x  +  4  =  0 
and  2  is  the  only  root ;  when  A:  =  —  1,  the  equation  is 

a;  +  4a;  +  4  =  0 
and  —  2  is  the  only  root. 

22.  x'-Q,kx  +  12  =  0.  27.    4a;2-|-4A:aj+4A;2=3(a;-l). 

23.  ^-kx-^l  =  0.  28.    k''x'-\-2kx  +  l  =  2x, 

24.  10a^  +  6;i^.  +  ^^-4  =  0.        ^'-    (^-  +  3)^  =  4(.  +  2). 

30.    {2x  +  ky  =  'd{x-^). 

25.  x^ +  2kx-\-k'^  =  ^x.  ,,/     ,   nxo        2 

31   ^  (y  +  i)   I  ?/^i 

26.  9a;2-f  6A:a;  +  A;2  =  10a;.  '  9  16 

32.    3  aj2 4- 5  A;a;(a;  -  5)  -  2  A;2(aj -  5)^  +  5  x  +  3  A:(a;  -  5)  +4  =  0. 

72.  Graphical  solution  of  the  quadratic  equation.  —  Suppose 
we  draw  the  locus  or  graph  of  the  equation 

2/  =  a;2. 

The  first  thing  to  do  is  to  find  pairs  of  values  of  x  and  y  that 
satisfy  this  equation.  This  can  be  done  by  taking  values  of  x 
at  will  and  computing  the  corresponding  values  of  y.  Some  of 
the  results  are  given  in  the  following  table : 


X 

-5 

-4 

-3 

-2 

-  1 

-i    0 

1 

2      1 

3      4      5 

y 

25 

16 

9 

4 

1 

I      0 

1 

4   H 

9     16    25 

If  we  plot  the  points  whose  coordinates  are  these  corresponding  pairs 
of  values  of  x  and  y  and  connect  these  points  by  a  smooth  curve,  we  shall 
have  an  approximate  representation  of  the  locus. 

If  now  we  draw  the  locus  of  the  equation 

y  =  x-^2 

with  reference  to  the  same  set  of  axes,  the  two  loci  will  intersect  in  two 
points.     The  coordinates  of  these  two  points  must  satisfy  both  equations. 


94 


COLLEGE  ALGEBRA 


or 


and  hence  the  abscissae  are  values  of  x  that 
make  the  right  members  of  these  equations 
equal ;  that  is,  for  values  of  x  equal  to  these 

abscissae 

a;2  =  X  +  2, 

or  x2  -  x  -  2  =  0. 

These  abscissae  are  therefore  the  roots  of 
this  last  equation. 

These  considerations  suggest  a  gen- 
eral method  for  solving  a  quadratic 
equation  graphically.  If  the  equa- 
tion is 

9_      h         c 


we  draw  the  loci  of  the  tvro  equations 
and  y  = x 


and  measure  the  abscissae  of  their  points  of  intersection.    These 
abscissae  are  the  roots  of  the  given  equation. 

The  locus  of  the  first  equation  is  called  a  parabola.  The 
locus  of  the  second  equation  is  evidently  a  straight  line. 

If  the  line  and  the  parabola  have  two  points  in  common,  the 
quadratic  has  two  real  roots. 

If  the  line  and  the  parabola  have  only  one  point  in  common,  the 
quadratic  has  only  one  root. 

If  the  line  and  the  parabola  have  no  points  in  common,  the  roots 
of  the  quadratic  are  imagiiiary.  In  this  case  this  method  gives 
us  no  information  concerning  the  roots  beyond  the  fact  that 
they  are  imaginary. 

73.   Find  the  roots  of  the  equation 

a;2_3aj-f-7  =  0. 


QUADRATICS 


95 


We  have  the  locus  of  the  equation 


already  drawn.     If  we  draw  the  locus  of 

y  =  Sx-7, 

we  find  that  the  two  loci  have  no  points  in  common  and  we  conclude  that 
the  roots  of  the  quadratic  are  imaginary.  That  this  conclusion  is  correct 
can  be  verified  directly  by  consideration  of  the  value  of  the  discriminant 
of  the  given  equation. 


Find  the  roots  of  the  equation 

a;"'*  +  4  a?  +  4 


0. 


Using  the  same  figure  for  the  locus  of  the  equation 

y  =  ^\ 

and  drawing  the  locus  of  the  equation 

y  =  —  4  X  -  4, 

we  find  that  the  two  loci  have  only  one  point  in  common.     The  abscissa 
of  this  point  is  —  2.     Hence  —  2  is  the  only  root  of  the  given  equation. 

The   student  must  not  expect  to  get  exact  solutions  by   graphical 
methods. 

74.   Second  graphical  method.  —  There  is  another  graphical 
method  for  the  solution  of  a  quadratic  that  is  in  common  use. 
V    It  is  sufficiently  explaiued  by 
the  following  example. 

Suppose  that  we  wish  to  find 
the  roots  of  the  equation 

Draw  the  locus  of  the  equation 

y  =  y:^  —  X  —  2. 

In  order  to  do  this  we  take  values 
of  X  at  will  and  compute  from  the 
equation  the  corresponding  values  of 
y.    Then   the   smooth   curve   drawn 

through  the  points  whose  coordinates  are  these  corresponding  pairs  of 
values  of  x  and  y  is  the  approximate  locus  of  the  equation. 


96  COLLEGE  ALGEBRA 

K  this  locus  has  a  point  in  common  with  the  x-axis,  the  abscissa  of 
such  a  point  is  a  value  of  x  that  makes 

a;2  -  ic  -  2  =  0, 

that  is,  is  a  root  of  this  equation,  since  the  ordinate  of  every  point  on  the 
ar-axis  is  0. 

In  general,  in  order  to  sol^re  the  equation 

by  this  method,  plot  the  locus  of  the  equation 

y  =  ax^  -\-hx-\-  c, 

and  measure  the  abscissae  of  the  points  this  locus  has  in  com- 
mon with  the  ic-axis.  These  abscissae  are  the  roots  of  the  given 
equation. 

EXERCISES 
Solve  each  of  the  following  equations  graphically. 
Do  not  use  the  same  method  for  all  of  the  equations. 
1.  x^-^x-l  =  0.  11.   iB2-8a;  +  16  =  0. 


2.  ar^_a;-l  =  0.  12.  ic^  _  g  a;  +  20  =  0. 

\'^^   /  3.  2x'  +  x-2  =  0.  13.  a;2_8a;  +  6  =  0. 

^4.  ar  +  3a;-|-l  =  0.  14.  3a;2  =  0. 

5.  5a^  +  7a:-2  =  0.  15.  a:^^! 

6.  5x'-^-lx-^2  =  0.  16.  a.-2  +  5a;  +  l=0. 

7.  2a52  +  8a;  +  8  =  0.  17.  a;2  +  3a;  +  2  =  0. 

8.  iB2  +  6a?  +  10  =  0.  18.  («4-4)2=:9. 

9.  3a^  +  8a;  +  5  =  0.  19.  (3a;-8)2  =  0. 

10  8a^  +  3a;-5  =  0.  20.  3  aj^-f- 2  »  + 1  =0. 

75.  Equations  involving  fractions.  —  When  we  clear  an 
equation  of  fractions  by  multiplying  each  member  by  the 
lowest  common  denominator  of  the  fractions,  the  resulting 


QUADRATICS  97 

equation  may  have  roots  that  are  not  roots  of  the  original 
equation.  In  other  words,  the  new  equation  may  not  describe 
the  same  numbers  as  are  described  by  the  original  equation. 


Consider,  for  example,  the  equation- 

2x  1 


+  4. 


x2  _  9       X- 

Clearing  of  fractions,  2  a;  =  x  +  3  +  4(x2  -  9), 

(x_3)-4(rK2-9)  =  0, 
(a;_3)(l-4a;-12)  =  0. 

One  of  the  roots  of  this  equation  is  3.  But  this  evidently  cannot  be 
a  root  of  the  original  equation,  since  for  aj  =  3  the  left  member  becomes  ^, 
and  this  has  no  numerical  value. 

76.  In  many  cases  the  roots  of  the  new  equation  —  that  is, 
the  equation  resulting  from  clearing  the  original  equation  of 
fractions  —  are  the  same  as  the  roots  of  the  original  equation. 
Some  of  the  circumstances  under  which  this  can  occur  may  be 
inferred  from  the  following  theorem : 

Theorem.  —  Any  root  of  the  new  equation  that  is  not  a  root 
of  the  original  equation  must  when  substituted  for  x  make  some 
denominator  of  the  original  equation  zero. 

Suppose  that  the  equation  has  been  brought  to  the  form 

by  transposing  all  the  terms  of  the  right  member  to  the  left 
member.     Here  P  represents  some  fractional  expression. 

If  D  is  the  lowest  common  denominator  of  the  fractions  oc- 
curring in  P,  the  new  equation  is 

PZ)  =  0, 

and  PD  is  a  polynomial  in  x. 

If  now  any  value  of  x  makes  PD  equal  to  zero  and  D  different 
from  zero,  it  must  make  P  equal  to  zero.  That  is,  it  must  be 
a  root  of  the  original  equation.  Hence  any  root  of  the  new 
equation  that  is  not  a  root  of  the  original  equation  must  be  a 
root  of  the  equation  7>  =  0 


98  COLLEGE  ALGEBRA 

But  any  root  of  this  last  equation  must,  when  substituted 
for  X,  make  some  denominator  of  the  original  equation  zero. 
(See  10,  Chap.  I.) 

It  follows  from  this  that  when  no  denominator  of  the  origi- 
nal equation  contains  the  unknown,  the  roots  of  the  new  equa- 
tion must  be  the  same  as  those  of  the  original  equation ;  that 
is,  the  two  equations  must  be  equivalent. 

77.  Hence,  after  having  solved  the  new  equation,  the  student 
should  determine  by  trial  whether  any  of  its  roots,  when  sub- 
stituted for  X,  makes  any  denominator  of  the  original  equation 
equal  to  zero.  No  root  of  this  kind  is  a  root  of  the  original 
equation.  But  all  the  other  roots  of  the  new  equation  are 
roots  of  the  latter,  and  every  root  of  the  original  equation 
is  a  root  of  the  new  equation. 

If  we  clear  the  equation  of  fractions  by  multiplying  each 
member  by  a  polynomial  that  is  a  common  denominator  not  of 
the  lowest  degree,  the  two  equations  will  certainly  not  be 
equivalent. 

EXERCISES 

Solve  each  of  the  following  equations : 


2. 


3. 


11.  .02  aj2- 3.64  a; -2.5  =  0. 

12.  .01a;(.3a;  +  4.5)  =  .34a;2_5^ 

1       .1.1 
2(a;-3)~a;-2  "      x-k 


^^ 


QUADRATICS  99 

14.  M±.-^+l  =  l.  16.   l  +  ^_  + J_  =  «. 
ar^— 9      a;  —  3  a;i»  —  la?  —  2a; 

15.  ^-^  I  ^-H-g^ct  17    _3 ^  =  4. 

a;4-a     a;— a      6  a;  +  l      x  —  1 

18.   ^^ +  ^=1. 

x'-Sx  +  'J      x-2 

78.  Equations  involving  radicals.  —  Certain  equations  involv- 
ing radicals  can  be  solved  by  the  methods  described  in  the 
following  illustrative  examples : 

It  is  understood  that  in  these  problems  we  are  to  attach  to 
every  radical  its  principal  value. 

Example  1.     Solve  the  equation 


V:c'2  -  9  -  4  =  0. 
Transposing  —  4  to  the  right  member, 


Squaring  both  members, 


Check.  V25  -9-4  =  4-4  =  0. 


Vx'- 

-9 

=  4. 

x^- 

-9 

=  16, 

X2 

=  25, 

X 

=  ±5. 

In  dealing  with  rational  integral  equations  we  checked  our 
results  merely  for  the  sake  of  guarding  against  error.  But  in 
solving  an  equation  with  radicals  we  sometimes  get  an  ap- 
parent root  that  is  in  reality  not  a  root  even  when  we  have 
proceeded  in  the  regular  way.  This  also  happened  in  the 
solution  of  fractional  equations.  Hence  the  operation  of 
checking  is  a  necessary  part  of  the  solution  of  such  equations. 
This  is  illustrated  in  the  next  example. 

Example  2.     Solve  the  equation 

Vx^=^  +  4  =  0. 
Transposing,  Vx  —  6  =—4. 

Squaring  both  members,  cc  —  6  =  16, 

X  =  22. 


100  COLLEGE  ALGEBRA 

Check.  V22-6  +  4  =  Vl6  +  4  =  4+4=^  0. 

Hence  22  is  not  a  root  of  the  equation.     As  a  matter  ot  fact,  the  equa- 
tion has  no  root. 

The  explanation  of  the  appearance  of  these  false  roots  is  as 
follows : 

If  the  original  equation  is  L  =  Ej 
the  resulting  equation  is  1/  =  R^, 
or  L^-li^  =  0. 

The  roots  of  this  last  equation  are  the  roots  of  the  equations 
L-Ii  =  0 
and  L-\-E  =  0. 

And  if  the  equation      L-\-  R  =  0 

has  a  root,  this  is  also  a  root  of  the  equation 

that  is  not  a  root  of  the  equation 

L  =  R, 
provided  that  it  does  not  make  L  =  0  and  /?  =  0. 


In  example  (1)  L  =  Vx^  —  9  and  jf2  =  4.  Here  the  equation 
iy  +  i2  =  0  has  no  root,  and  therefore  the  equation  U  =  R^  is 
equivalent  to  the  equation  L=R. 

But  in  example  (2)  7^  =  Vic  — 6  and  i2  =  — 4,  and  therefore 
the  equation  i  +  i?  =  0  has  a  root ;  namely,  22.  This  is  not  a 
root  of  the  original  equation,  since  in  this  case  R^O  for  any 
value  of  X. 

Example  3.     Solve  the  equation 


Vx+l  +  Vx^^  =  5. 


Transposing  y/x  —  3  to  the  right  member, 


Vx+  l=5-\/a;-3. 
Squaring  both  members,        a;  +  1  =  25  —  lOVa;  -  3  +  a;  —  3. 


QUADRATICS  101 


Transposing,  lOVx  —  3  =  21. 

Squaring  both  members, 

100(x  -  3)  =  441, 
x  =  7.41. 


Check.         V7.41  -f  1  +  \/7.41  -  3  =  V8:41  +  ViAl  =  2.9  +  2.1  =  5. 
Example  4.     Solve  the  equation 


Vx2  -  10  X  +  41  -  Vx2  -f  10  X  +  41  =  8. 


Transposing  —  Vx^  +  10  x  +  41  to  the  right  member, 


Vx2  _  10  X  4-  41  =  8  +  Vx-2  +  10  X  +  41. 


Squaring  both  members, 


x2  _  10  X  +  41  =  64  +  16  Vx'^  +  10  X  +  41  4-  a;2  +  10  X  +  41, 
-20x-64  =  16Vx2  +  lOx  +  41, 


5x  +  16  =-4Vx2 +  10x4-41. 

Squaring  both  members, 

25  x2  +  160  X  +  256  =  16  x-2  +  160  x  +  656,     • 

9  x2  =  400, 

x  =  ±^. 
Check  for  x=  \^-. 

^y±o(i.  _  ioo.  +  41  _  Vioo  _,.  20  0  _(.  41  ^  Vi|^_  VI^^  J/  -  -V-  =-8. 

Hence,  ^^-  is  not  a  root  of  the  equation. 

Check  for  x  =  —  ^^. 

V^lM^ipTri  -  V^^  _  Aoo  _,.  41  =  5^  -  J/  =  8. 
Hence,  —  %"  is  a  root  of  the  equation. 

79.  The  preceding  equations  contain  no  radicals  except 
square  roots.  The  most  common  irrational  equations  are  of 
this  kind  and  the  method  of  solution  illustrated  in  these  ex- 
amples is  usually  applicable  in  such  cases.  It  may  be  formu- 
lated as  follows : 

If  necessary,  so  transpose  the  terms  that  one  radical  stands 
alone  in  one  member  of  the  equation. 


102  COLLEGE  ALGEBRA 

Then  form  a  new  equation  by  equating  the  squares  of  the  menv- 
hers  of  the  last  equation. 

If  the  resulting  equation  contains  radicals,  repeat  this  process 
until  an  equation  is  obtained  that  is  free  from  radicals. 

Solve  this  last  equation  and  test  each  of  its  roots  by  substitut- 
ing it  in  the  original  equation. 

In  simplifying  the  radicals  in  order  to  determine  whether 
a  number  satisfies  the  equation,  the  student  must  not  resort 
to  squaring  both  members  of  the  equation,  since  the  question 
he  is  trying  to  settle  is  whether  the  equation  formed  in  this 
way  is,  or  is  not,  equivalent  to  the  original  equation. 

EXERCISES 
Solve  the  following  equations  and  check  your  results : 

1.    Va? -f •  2  — .'»  =  0.  ^     Va;^-|-  9      a;-5_17 


3.    V4  a;  + 13  +  a;  =  2.  5.    ■\/x'' ■\-^x-\-2  =  x  +  l. 


^     Va^2  +  16^a;-9 

t>-    -, 1 z, — 


=\  K^ 


\n 


7.  V.t2  -f  10  a;  +  41  -  Vif2  -  10  a;  +  41  =  8. 

, , \ 

8.  Vcc2  +  5i»  +  22-Va;2-5a;4-22==2.  ^    ^      \ 

S,' 

10.    V(a;  +  2)2  +  3+V(a;-l)2  +  3  =  5.  '^ 


11.    V8x  +  1  =  vV  +  6 a;  —  2. 
^12.    V(aJ-3)2  +  7+V(aj-|-7)2  +  7=8. 


'^  13.    V5a;-6-Va;-2  =  2. 


14.    V3a;4-10  =  Va;-l+V2a;-l. 


QUADRATICS  103 


15.  1125  =  1093  VI  +  .003665  t 


16.  1?  =  1093  VI +  .003665^. 

Solve  for  t.  This  formula  gives  the  velocity  of  sound  in  air  in  feet  per 
second  at  a  temperature  of  t°  Centigrade.  The  formula  in  Ex.  15  is  a 
particular  case  of  this. 

17.  V5  +  (6  -  xy  =  5. 


18.    \/9  +  (4  +  a?)2  =  V(3  +  xf  +  16. 


19.  x  —  l—Wl  +  x  =  0. 

20.  2x  +  3  =  4-V^. 

21.  ■\/{x  _  3)2  +7  -  ^{x  +  7)2  +  7  =  8. 


22.    v/2/2-2/  +  l+V2/2-22/+l  =  l. 

80.  Equations  in  the  form  of  quadratics.  —  We  can  frequently 
apply  the  methods  for  solving  quadratics  to  the  solution  of 
equations  that  involve  the  unknown  only  in  a  certain  expres- 
sion and  in  the  square  of  this  expression. 

Example  1.     Solve  the  equation 

3  ic*  -  14  a;2  4-  8  =  0. 
This  can  be  written     3  {x^Y  -  14  x2  +  g  =  0. 


Hence,  _  14  ±  \/l96  -  96 


=  4,  orf, 


6  '       3 

X  =  ±  2,  or  ±  Vf. 

The  student  should  check  these  results. 
Example  2.     Solve  the  equation 


3  ic  -  5  -  4  \/3x-5  +  3  =  0. 
This  can  be  written 


CV3a:-5)2  _4V3a:-5  +  3  =  0. 
Hence,  V3a:  -  6  =  3,  or  1, 

3  X  —  5  =  9,  or  1, 
X  =  ^5*,  or  2. 
The  student  should  check  these  results. 


104  COLLEGE  ALGEBRA 

Example  3.     Solve  the  equation 


x2  -  s  X  -\-  5  -  S  y/x^  -  S  X  +  S  =  -  5. 


This  can  be  changed  into  a  quadratic  in  Vx^  —  3  x  +  8  by  adding  3  to 
each  member.     Thus, 


x2  -  S  X  +  8  -  S  Vx^  -  S  X  +  8  =-  2. 
Hence,  \/x2  -  3  a;  +  8  =  2,  or  1, 

a;2__3a;4-8  =  4,  or  1. 

3±VHI,,,3±VEI2 
2         '  2 

The  student  should  check  these  results. 


EXERCISES 

Solve  the  following  equations  : 


4.    -6Va;2+l  +  a;2  +  lO  =  0.  (7;)  («2 _  1)24.  3(a?2  _  i) _^  2  =  0. 


9.   a^  +  26a^-27  =  0. 

10.  a;*- 2x2  +  1  =  0. 

11.  2/' -13  2/^^  +  36  =  0. 


12.  7Va;2  +  3a;  +  2x24-6a;-4  =  0. 

13.  a;2  +  8a;  +  l  +  3Va;2  +  8a;  +  2  =  9. 

{< 

.      -.15.    (z'-j-Szy-2(z^  +  3z)-S  =  0. 

16.  2:2^-1122  +  12  =  0.  18.    (2/2-5)2  =  16. 

17.  a;«  +  7ic3-8  =  0.  19.    -6 Va;  +  l  +  a;  +  10  =  0. 


14.    lx^±]'-5fx-{--]  +  6=:0. 

X. 


''■  {h-'J-ih-'^'^'- 


QUADRATICS  105 

81.  The  conditions  of  many  problems  require  that  the  un- 
known number  be  real.  If  such  a  problem  leads  to  a  quadratic 
equation  and  if  one  of  the  known  numbers  is  represented  by  a 
letter,  we  can  frequently  find  limitations  on  the  possible  values 
of  this  known  number.  The  method  of  procedure  is  illustrated 
in  the  following  example. 

Find  two  positive  numbers  such  that  their  product  is  36  and 
their  sum  s. 


Let 

X  =  one  of  the  numbers. 

Then 

s  —  a:  =  the  other  one. 

Hence 

x(s-x)=  36, 

or 

x^ 

-  sx  +  36  =  0. 

In  order  for  the  roots  of  this  equation  to  be  real  we  must  have  s^  —  144 
^  0.     Hence  the  least  possible  value  for  s  is  12. 

What  are  the  required  numbers  for  this  value  of  s  ? 


PROBLEMS 

In  solving  a  quadratic  it  is  generally  best  to  use  the  formula, 
unless  the  method  of  factoring  can  be  applied  readily. 

Formula  from  physics.  —  The  number  n  of  complete  vibrations 
per  second  made  by  a  stretched  wire  is  given  by  the  formula 


1     / 
=  21^1 


9S0M 


where  I  is  the  length  in  centi-      j^  2^ 

meters  between  the  bridges,  M      I 

the   weight   in    grams   of    the     ^ 

stretching  weight,  and  m  the  weight  in  grams  of  the  wire  per 

centimeter  of  length. 

1.  What  must  be  the  stretching  weight  on  a  wire  58.6 
centimeters  long  whose  weight  per  centimeter  is  .0098  gram  in 
order  that  it  may  make  256  complete  vibrations  per  second  ? 


106  COLLEGE  ALGEBRA 

Formula  from  physics.  —  The  velocity  v  of  a  projectile  at  any 
moment  is  given  by  the  formula 


V  =  W  —  64  2/ 

where  Vq  is  the  initial  velocity  and  y  is  the  height  of  the  pro- 
jectile at  this  moment  above  the  level  of  the  starting  point. 

2.  A  projectile  is  fired  from  the  ground  with  an  initial 
velocity  of  225  feet  per  second.  How  high  will  it  be  when  it 
has  a  velocity  of  150  feet  per  second  ? 

3.  Let  P  be  a  point  on  a  semicircle  whose  diameter  is  ABj 
and  let  PB  be  the  perpendicular  from  P  to  AB.     Being  given 

^ -^^^  that  AB  =  10  inches,  find  the  length  of 

^^  ^\  EP  Yfhen  AR-^PP  =7^  mohes.    (PP 

I  \  is   a   mean   proportional   between  AR 

^lI 1^  and  RB.) 

4.  Given  the  points  P  and  Q  with  the  coordinates 
(V3,  —2)  and  (V3,  1)  respectively.  Find  the  point  A  on 
the  2/-axis  such  that  PA-{-AQ  =  6. 

5.  Find  the  coordinates  of  all  the  points  on  the  line 
y  =  V2,  the  sum  of  whose  distances  from  the  points  (2,  0) 
and  (—2,  0)  is  equal  to  5. 

6.  Find  the  coordinates  of  the  points  on  the  line  x  =  3  that 
are  10  units  from  the  point  (9,  2). 

7.  A  box  is  to  be  3  feet  long  and  1  foot  wide.  How  deep 
must  it  be  in  order  to  have  a  diagonal  of  4  feet  ? 

8.  The  points  A  and  B  on  the  avaxis  have  the  abscissae  4 
and  12  respectively.  What  must  be  the  abscissa  of  the  point 
C  on  the  aj-axis  in  order  that  the  area  of  the  circle  with  C  as  a 
center  and  passing  through  A  shall  be  twice  that  of  the  circle 
with  (7  as  a  center  and  passing  through  B?  Explain  geomet- 
rically why  there  should  be  two  answers. 

9.  What  must  be  the  abscissa  of  C  in  order  that  the  area  of 
the  first  circle  of  Problem  8  be  one  fourth  that  of  the  other  one  ? 


QUADRATICS  107 

10.  The  points  A  and  B  on  the  a;-axis  have  the  abscissae 
—  3  and  8  respectively.  What  must  be  the  radii  of  the  two 
circles  with  centers  at  A  and  jB  respectively,  and  tangent  to 
each  other  in  order  that  the  area  of  the  former  shall  be  five 
times  that  of  the  latter?  Explain  geometrically  why  there 
should  be  two  answers. 

11.  The  points  A  and  B  on  the  a>axis  have  the  abscissae  a 
and  b  respectively.  What  must  be  the  radii  of  two  circles  with 
centers  at  A  and  B  respectively,  and  tangent  to  each  other  in 
order  that  the  area  of  the  former  shall  be  k  times  that  of  the 
latter  ? 

12.  Find  the  abscissae  of  the  points  in  which  the  circle 

(x-2y  +  (y-hSy  =  25 
cuts  the  £P-axis.     Draw  the  figure. 

13.  Eind  the  abscissae  of  the  points  in  which  the  circle 

(x-f  2)2 +(2/ 4- 3)2  =9 
cuts  the  a>axis.     Draw  the  figure. 

14.  Where  does  the  circle 

(,,  +  5)2 +  (2/ -6)2  =  16 
cut  the  2/-axis  ?     Draw  the  figure. 

15.  What  must  be  the  dimensions  of  a  rectangular  field  that 
is  to  contain  3  acres  and  have  a  perimeter  of  100  rods  ? 

16.  What  is  the  least  possible  perimeter  of  a  rectangular 
field  that  is  to  contain  10  acres  ? 

17.  What  is  the  least  possible  perimeter  of  a  rectangular 
field  that  is  to  contain  20  acres  ? 

18.  What  is  the  greatest  possible  area  of  a  rectangular  field 
inclosed  by  200  rods  of  fence  ? 

19.  What  is  the  area  of  the  greatest  rectangle  that  can  be 
inclosed  by  a  rope  60  feet  long  ? 

20.  A  man  is  in  a  boat  3  miles  from  the  nearest  point  of 
a  straight  beach.     If  he  rows  at  the  rate  of  4  miles  an  hour 


108  COLLEGE  ALGEBRA 

and  walks  at  the  rate  of  5  miles  an  hour,  where  ought  he  to 
land  in  order  to  reach  a  point  on  the  beach  5  miles  from  the 
point  directly  opposite  him  in  one  hour  and  twenty-seven 
minutes  ? 

Would  it  be  possible  for  him  to  reach  his  destination  in  one 
hour? 

21.  An  open  box  with  a  square  base  and  a  volume  of  392 
cubic  inches  is  to  be  made  by  cutting  2  inch  squares  from  the 
corners  of  a  piece  of  tin  and  turning  up  the  sides.  How  large 
a  piece  of  tin  should  be  used  ? 

22.  A  man  and  a  boy  working  together  do  a  piece  of  work 
in  8  days.  If  we  assume  that  the  man  could  do  it  alone  in  12 
days  less  than  the  boy  could  do  it,  how  long  would  it  take  each 
to  do  it  ? 

23.  The  sides  of  two  rectangles,  one  of  which  is  within  the 
other,  are  parallel  and  equal  distances  apart.  How  far  apart 
are  the  sides  if  the  outer  rectangle  is  10  centimeters  by  14 
centimeters  and  is  twice  as  large  as  the  inner  one  ? 

Note.  — The  equation  obtained  in  the  solution  of  a  problem  should  ex- 
press all  the  conditions  of  the  problem  if  possible.  But  in  some  prob- 
lems there  are  conditions  that  cannot  be  expressed  in  the  equation.  In 
Problem  23,  for  example,  the  implied  condition  that  the  number  sought 
must  be  less  than  one  half  the  smaller  dimension  of  the  outer  rectangle 
cannot  be  expressed  in  the  equation.  Hence  we  must  apply  this  condition 
to  the  roots  after  the  equation  has  been  solved  and  reject  any  that  do  not 
satisfy  it. 

24.  Two  men  rowed  12  miles  upstream  and  back  in  7  hours 
and  30  minutes.  The  current  was  running  at  the  rate  of  3  miles 
an  hour.  How  fast  would  they  have  gone  in  still  water  ?  Ex- 
plain the  two  solutions. 

25.  A  lever  is  to  be  cut  from  a  bar  weighing  3J  pounds  to 
the  foot.  How  long  must  it  be  in  order  that  it  may  balance 
about  a  point  2  feet  from  one  end  when  a  weight  of  16  pounds 
is  attached  to  this  end  ?    Explain  the  two  solutions. 


QUADRATICS  109 

26.  A  stone  is  dropped  over  the  edge  of  a  cliff  and  12  sec- 
onds later  the  sound  of  its  striking  the  ground  below  is  heard 
on  the  cliff.  How  high  is  the  cliff  if  the  velocity  of  sound  is 
taken  as  1120  feet  per  second  ? 

27.  Find  the  outer  radius  of  a  hollow  spherical  shell  an  inch 
thick  whose  volume  is  125  cubic  inches. 

28.  Find  the  outer  edge  of  a  hollow  cubical  shell  an  inch 
thick  whose  volume  is  56  cubic  inches. 

29.  The  two  legs  of  a  right  triangle  are  3  inches  and  4  inches 
long  respectively.  If  the  shorter  one  remain  unchanged,  by 
how  much  will  the  longer  one  need  to  be  lengthened  in  order 
to  lengthen  the  hypotenuse  2  inches  ? 

30.  If  the  longer  leg  of  the  right  triangle  of  Ex.  29  remains 
unchanged,  by  how  much  will  the  shorter  one  need  to  be 
lengthened  in  order  to  lengthen  the  hypotenuse  2  inches  ? 

Center    of      gravity.  —  Let         ^      ^       ^ 
A  and  B  be  two  bodies  of      '  C    ^     '^  a  ,?„- 

weights    Wi   and    W2  re-  ^ 

spectively,  and  let  the  distances  of  their  centers  of  gravity 
from  the  point  0  in  a  line  with  A  and  5  be  a  and  b  respec- 
tively. Then,  as  explained  in  physics,  the  distance  from  0  of 
the  center  of  gravity  of  the  two  together  is 

31.  A  tin  can  open  at  the  top  has  a  diameter  of  4  inches 
and  a  height  of  6  inches.  It  weighs  4  ounces  and  its  center  of 
gravity  is  2^  inches  above  the  center  of  its  base.  How  much 
water  must  be  poured  into  it  in  order  to  bring  the  center  of 
gravity  down  to  21  inches.  A  cubic  inch  of  water  weighs  f  J 
of  an  ounce.     Explain  the  two  solutions. 


CHAPTER   VIII 


SYSTEMS   OF   EQUATIONS   IN   TWO   UNKNOWNS 
SOLVABLE   BY   MEANS    OF    QUADRATICS 

82.    Case  I.     One  of  the  equations  is  linear  and  the  other  one 
is  a  quadratic;  that  is,  of  the  second  degree. 

In  this  case  we  can  proceed  as  illustrated  in  the  following 
example. 

Solve  the  equations  x^  +  y^  =  16,  (1) 

x  +  y  =  l.  (2) 


From  (2)  we  get  y  =  l  —  x. 

Substituting  this  value  of  y  for  y  in  (1),  we  get 


x2^(l-x) 


(3) 


2x2 


Then  from  (3) 


Hence  the  pairs  of  values  of  x  and  y  that  satisfy  both  equations  are 

..    i-VsI 


y  = 

The  student  is  familiar  with  the 
fact  that  any  linear  equation  in  x 
and  2/  has  for  its  locus  a  straight 
line.  He  also  knows  that  the  locus 
of  equation  (1)  is  a  circle  with  its 
center  at  the  origin  and  radius  equal 
to  4.  Since  the  pairs  of  values  of  x 
and  y  just  found  satisfy  both  equa- 

110 


SYSTEMS  OF  EQUATIONS  IN  TWO  UNKNOWNS      111 

tions,  they  are  the  coordinates  of  the  points  of  intersection  of 
this  circle  and  the  straight  line  represented  by  equation  (2). 

If  only  one  pair  of  values  of  x  and  y  satisfy  both  equations, 
the  straight  line  is  tangent  to  the  curve.  If  the  values  of  x 
and  y  that  satisfy  both  equations  are  imaginary,  the  line  does 
not  meet  the  curve.  Conversely,  if  the  line  does  not  meet  the 
curve,  we  know  that  the  values  of  x  and  y  which  satisfy  both 
equations  are  imaginary,  but  we  do  not  know  from  this  fact 
their  particular  values. 

If  the  equations  are  satisfied  by  real  values  of  x  and  i/,  these 
values  can  be  determined  graphically  by  drawing  the  locus  of 
each  equation  and  measuring  the  coordinates  of  their  common 
points. 

EXERCISES 

Solve  each  of  the  following  pairs  of  equations.  Check  your 
work  by  means  of  graphs.  The  loci  of  the  first  equations  in 
Exs.  11  and  12  are  circles  whose  centers  are  not  at  the  origin. 
Do  not  attempt  to  draw  these  loci. 

r  1.   aj2  +  2/2  =  9,  -^     7.   ic2  +  2/'  =  40, 

\  2.    3?-\-y'^=^,  \l  8.    x  +  y  =  0, 

^     2  a;  +  2/  =  5.  a^  +  2/^  =  35. 

"3.   ic2  +  jy'  =  16,  9.   2x  =  by, 

Sx-2y  =  0.  x'  +  y^  =  12. 

4.  a;2 -1-2/2  =  1,  10.    x^  =  10-y% 
3x-\-4:y-\-l  =  0.  y  =  x-10. 

5.  x''+2f  =  10,  11.   a;2-}-2/2-4aj-6  2/  +  9=0, 
y  =  x-l.  y=x-^2. 

6.  a;2  +  2/2  =  25,  12.   ar^  +  Z- 6  a;-f  5  2/  =  2, 
2x-{-4:y  =  S.  3a;  +  22/  =  l. 


112 


COLLEGE  ALGEBRA 


13.  2/2  =  6-a;2, 

14.  i^-9  =  -y^, 
2x-\-y  +  l  =  0, 

15.  a;2  + 2/2  =  1, 
x-\-y  =  l. 

16.  a;2_|_2/2_9  =  0, 

y  =  2. 


17.  ^2  +  2/2- 9  =  0, 

18.  i»2  +  ?/  =  9, 
a;  =  4. 

19.  2/-l  =  3(a;  +  l), 

iK2  _|_  2/2  =  3. 

20.  2/  +  2=4(a:  +  3), 


83.  Equation  of  an  ellipse.  —  Not  all  equations  of  the  second 
degree  in  x  and  y  represent  circles.  We  have  already  met 
with  equations  of  this  kind  that  represent  parabolas  (see  §  72), 
and  there  are  equations  of  this  kind  that  represent  other 
curves. 

Consider,  for  example,  the  equation 


The  student  should  verify  that  the  following  pairs  of  values  of  x  and  y 
satisfy  this  equation : 


a? 

0 

0      3-8             1 

1 

2 

2         -1 

-1         -2 

-2 

y 

2 

-2      0      0      %V2 

-§\/2 

§V5 

-§\/5      §V2 

-§\/2    §V5 

-SV5 

Moreover,  x  =  ±  |-n/4  —  y^  and  y  =  ±  |  V9  —  o;^.  Hence  if  the  abso- 
lute value  of  y  exceeds  2,  a;  is  imaginary  ;  and  if  the  absolute  value  of  x 
exceeds  3,  y  is  imaginary.  This  means  that 
there  is  no  point  on  the  locus  whose  ordinate 
exceeds  2  in  absolute  value  and  no  point  whose 
abscissa  exceeds  3  in  absolute  value. 

If  we  plot  the  points  whose  coordinates  are 
the  pairs  of  values  of  x  and  y  in  the  foregoing 
table  and  connect  these  points  by  a  smooth 
curve,  we  shall  have  the  graph  of  the  equation. 

This  curve  is  called  an  ellipse. 


SYSTEMS  OF  EQUATIONS  IN  TWO  UNKNOWNS      113 


84.   Equation  of  a  hyperbola.  —  We  consider  next  the  equa- 


tion 


9 


4 


Solving  for  y,  we  get  ?/ =  ±  fVic^  —  9. 
Hence  there  is  no  point  on  the  locus  whose 
abscissa  is  less  than  3  in  absolute  value. 
Solving  for  x,  we  get  x  =±l Vy-  +  4. 
Hence  there  is  no  limitation  on  the  ordi- 
nates  of  the  points  of  the  locus, 

A  table  of  corresponding  values  of  x  and  y  is  as  follows 


X    3 

-3       4 

4           -4 

-4       5       5-5-5       6 

6           -  6 

-6 

V    0 

0   §V7 

-SVT    §V7 

-§^  §    -i    I    -s  -iV^ 

-   Vb"   2Vu 

-2V3 

This  curve  is  called  a  hyperbola. 

Curves  whose  equations  are  of  the 
form  xy  =  k  are  also  hyperbolas,  but 
they  are  situated  differently  with  re- 
spect to  the  coordinate  axes.  The  fig- 
ure gives  the  graph  of  the  equation 

xy=l. 

When  k  is  positive,  the  two  branches  of 
the  curve  are  in  the  first  and  third  quadrants  ;  when  k  is  negative,  the  two 
branches  are  in  the  second  and  fourth  quadrants. 

85.  Equation  of  a  parabola.  — The  graph  of  the  equation 

y'  =  Sx 

is  given  in  the  figure.  The  curve  is  a  parab- 
ola, but  situated  differently  with  respect  to 
the  coordinate  axes  from  the  parabola  men- 
tioned in  §  72.   C^-.  y^    UZ  / 

86.  Equation  of  two  straight  lines.  —  If 

the  equation  is  such  that  when  the  right 

member  is  zero  the  left  member  can  be  factored  into  two  factors 

of  the  first  degree,  the  locus  is  composed  of  two  straight  lines. 


114  COLLEGE  ALGEBRA 

Consider,  for  example,  the  equation 

Transposing  and  factoring,  we  get 

(x+2?/  +  3)(3x-y  +  2)=0. 

Now  any  pair  of  values  of  x  and  y  that  make  either  of  these  factors 
zero  must  satisfy  the  equation  and  are  therefore  the  coordinates  of  a 
point  on  the  locus.  And  conversely  the  coordinates  of  any  point  on  the 
locus  when  substituted  for  x  and  y  respectively  must  make  the  product 
of  these  factors  zero  (10,  Chap.  I).  Hence  the  locus  of  the  given 
equation  is  the  same  as  the  loci  of  the  two  equations 

a;  +  2  ?/  +  3  =  0, 
Sx-y  +  2  =  0. 

87.  •  These  are  the  simpler  forms  of  the  equations  of  the 
second  degree  in  two  variables.  But  there  are  many  others. 
The  equations  of  circles,  ellipses,  hyperbolas,  parabolas,  and  of 
two  straight  lines  are  of  the  second  degree. 

EXERCISES 

Solve  each  of  the  following  pairs  of  simultaneous  equations. 
(Check  your  work  by  means  of  graphs?) 


,v 


1. 

S-f-'. 

h 

5. 

y  =  4:X-\-l, 

y^-Sx  =  0. 

2/  =  3aj  +  l. 

k- 

6. 

xy=l, 

2. 

xy  =  -2, 

V 

4:X-2y  +  S  =  0. 

2xH-2/  +  l  =  0. 

7. 

x'-^6xy-h9f  =  - 

3. 

9~4~   ' 

^  +  ^  =  1. 
3^4 

2/  +  3=K^  +  l)- 

8. 

9+16"^' 

4. 

f  =  6x, 

/ 

4     5 

SYSTEMS  OF  EQUATIONS  IN  TWO  UNKNOWNS      115 


16       ' 


9.    x" 

3a;  +  2?/  =  0. 


10.  /  +  a;  =  0, 

11.  3x2/4-4  =  0, 


M^^ 


0. 


12.  4  2/^  —  5  a;  =  0, 
i/-2  =  3(x  +  l). 

13.  4ic^  +  5a'2/  +  a:  =  0, 
2aj-2/  +  4  =  0. 


14.    ^  +  ^' 
3      2 


1, 


a;  +  2/+l  =  0. 


15    ^_l!-i 

2a;  +  52/-fl  =  0. 

IS.    f  +  |=l, 

^--^  =  1 

9     16 

17.  ar^-2/2  =  0, 

3  a;  -  2  2/  =  4. 

18.  xy  —  5  =  0, 
y  =  2x-l. 

19.  7a^  — 3a;2/  +  2a;  =  0, 
5a;  +  2/  +  2  =  0. 


20.    — - 


^_2^ 
9      4 


0, 


0. 


21.   x^  —  5y  =  0, 
x-y  =  l. 


^\^^  88.  It  is  evident  that  the  intersections 
of  the  line  2/  =  3  a;  +  &  and  the  circle  r*  +  y^ 
— 10  depend  upon  the  value  of  6,  since 
this  value  determines  the  position  of  the 
locus  of  the  equation 

y  =  Zx-\-h. 

If  we  eliminate  y  between  the  two  equations, 
we  get 

a;2+(3x  +  6)2=10, 
10  a:2  +  6  6a:  +  V^  -  10  =  0. 
If  now  h  has  a  value  that  makes 
36  62_40(52_io)>o, 


116  COLLEGE  ALGEBRA 

the  roots  of  this  equation  are  real  and  distinct  and  therefore  the  line  cuts 
the  circle  in  two  points.     But  if  h  has  a  value  that  makes 
36  62  _  40(&2  _  10)  =  0, 

this  equation  has  only  one  root  and  the  line  is  tangent  to  the  circle.    If 
finally  h  has  a  value  that  makes 

36  62  _  40(62  _  io)<  0, 

this  equation  has  imaginary  roots  and  the  line  has  no  points  in  common 
with  the  circle. 

Now  36  62  -  40(62  _  10)  =  -  4  6^  +  400. 

n  then  -  4  62  +  400  >  0, 

or  62  <  100,  (See  §  182.) 

the  line  cuts  the  circle.     But  in  order  that  62  be  less  than  100,  6  must 
have  a  value  between  -  10  and  10  ;  that  is,  —  10  <  6  <  10.     If 

-  4  62  +  400  =  0, 

6  =±10, 
and  if  -  4  62  +  400  <  0, 

then  62  ^  loo, 

and  either  6  <  —  10  or  6  >  10. 

The  foregoing  results  are  summarized  in  the  following  table  : 


-  10  <  6  <  10 

Line  cuts  circle  in  two  points. 

6  =  -  10,  or  6  =  10 

Line  is  tangent  to  circle. 

6  <-  10,  or  6  >  10 

Line  has  no  point  in  common  with  circle. 

Definition.  —  A  literal  constant  that  occurs  in  the  equation 
of  a  curve  is  called  a  parameter. 

Thus  in  the  equation  ?/  =  3  ic  +  6,  6  is  a  parameter. 

If  the  equation  of  a  straight  line  contains  a  parameter,  we 
can  frequently  determine,  as  in  the  example  just  given,  the 
values  of  the  parameter  for  which  the  line  cuts  the  locus  of  a 
given  equation  of  the  second  degree  in  two  points ;  the  values 
of  the  parameter  for  which  the  line  has  only  one  point  in  com- 
mon with  this  locus ;  and  the  values  of  the  parameter  for  which 
the  line  has  no  points  in  common  with  this  locus. 


SYSTEMS  OF  EQUATIONS  IN  TWO   UNKNOWNS    117 

EXERCISES 

Solve  each,  of  the  following  pairs  of  equations.  Find  the 
values  of  the  parameter  for  which  the  line  has  but  one  point 
in  common  with  the  curve.  Find  also  at  least  one  value  of  the 
parameter  for  which  the  line  cuts  the  curve  in  two  points,  and 
at  least  one  value  for  which  the  line  and  the  curve  have  no 
common  point. 


1. 

a;2  +  2/2  =  16, 

A 

!-.-. 

11. 

-x2  +  2/^  =  16, 

y  =  mx-\-  5. 

o» 

y  =  2x-\-h. 

y  =  3x-{-b. 

^J' 

^  +  /  =  16, 

7. 

if                                    • 

9      16~    ' 

12. 

xy  =  l, 
x-^y  =  k. 

3. 

^  +  f  =  l, 

x-\-y  =  a. 

13. 

y'=:3x, 

|  +  -^  =  1. 

8. 

a,'2-|-/  =  49, 

y  =  mx  -f  1. 

2      m 

y  =  mx  -\-  7. 

4. 

^  +  f  =  % 

9+16"^^ 

14. 

xy  +  4.  =  0, 

y—4:=m(x  —  5). 

9. 

y  =  mx. 

5. 

f'+.^=i, 

10. 

y  =  h. 
xy  =  l, 

15. 

^  +  t  =  o, 

4       9        ' 

y  =  3x-{-b. 

y  =  2x  +  h. 

y  =  mx-{-3. 

89.  Case  11.  Each  equation  contains  each  unknown  only  to 
the  second  power;  that  iSj  there  are  no  terms  of  the  first  degree  and 
no  terms  containing  xy. 

In  this  case  we  first  solve  the  equations  for  a?  and  /  and 
then  from  these  values  get  x  and  y.  The  general  procedure  is 
illustrated  in  the  following  example : 

Solve  the  equations  x^  +  y'^  =  16,  (1) 

^+1^=1.  (2) 

25      9 

Eliminating  y"^,  we  get  16  x^  =  175, 


118 


COLLEGE  ALGEBRA 


Eliminating  x  and  solving  for  y,  we  get 

By  substituting  these  values  of  x  and  y  for  x  and  y  respectively  in 
equations  (1)  and  (2),  we  see  that  either  value  of  x  can  go  with  either 
value  of  y. 
Thus, 

_5\/7     5  V7     -5V7      -5\/7. 

'^    ~A  '  A  1  ~A  '  ~A  ' 

4  4  4  4 


y  = 


Hence  there  are  four  solutions  to  these 
equations  and  the  loci  intersect  in  four  points. 


EXERCISES 


Solve  each  of  the  following  pairs  of  equations.  Check  your 
work  by  means  of  graphs.  Draw  the  graphs  of  the  equations 
in  Exs.  1-3  with  reference  to  the  same  set  of  axes ;  also  those 
in  Exs.  4-7 ;  and  those  in  Exs.  8,  9. 


1. 

0^^  +  2/^  =  16, 

6.   a;2  + 1/^  =  4,       .. 

'  HO. 

iE!  +  -^'  =  l,_. 

9      25 

\^-^^  +  t  =  l.     ' 
'            25  ^  9 

16^9*       ' 
-'^5      9 

ik  ' 

>i 

2. 

^  +  /  =  9, 
9      25 

.7.  a^'^y  =  36, 

11. 

x^  =  4.-f. 

3. 

9      25 

""^i+f-'. 

12. 
13. 

a^  +  2/'  =  25. 
2.-^2  +  32/2  =  60 

4. 

■^  +  ^'  =  1, 
25      9 

25      9 

5       4 
4       5 

ar'  +  2^  =  25. 

«•  ^  +  1  =  1' 

14. 

^^--1=1, 

5. 

^  +  f  =  % 

16      9 

9      4       ' 

^  +  1  =  1. 

^-^^  =  1. 

^  +  2/^  =  1. 

25      9 

25      9 

4      ^ 

SYSTEMS  OF  EQUATIONS  IN  TWO  UNKNOWNS      119 


15. 

^-/=o, 

17.    3)2  +  ^2^25,           19. 

4a:2==92/2, 

16. 

a:2  +  /=25, 
5      4 

5      4 

20. 

18.   3a:''^4-5/=32, 

.t2  =  2/^. 

40:2  +  92/^  =  1. 

0^^  +  2/^  =  1, 

90.  We  have  discussed  in  some  detail  the  two  simplest  cases 
that  can  arise  in  the  solution  of  sets  of  equations  in  two  un- 
knowns solvable  by  quadratics.  In  general,  the  solution  of 
a  set  of  two  equations  in  two  unknowns  is  beyond  the  scope  of 
this  book.  There  are,  however,  certain  ingenious  special  de- 
vices that  can  sometimes  be  used.  We  shall  explain  two  of 
these.  The  others  are  not  sufficiently  important  to  the  student 
to  justify  a  discussion  of  them  here. 

(a)  Solve  the  equations     x^  -\-y^  =  21^  (1) 

X  +  y  =  3.  (2) 

Equation  (1)  can  be  written 

(x  +  y)(x2-a-y  +  ?/2)  =  27.  (3) 

If,  in  the  left  member  of  (3)  we  substitute  iov  x  +  y  its  value  as  given 
by  (2),  we  get 

or  x^  +  xy  +  y'^  =  9.  (4) 

Now  (2)  and  (4)  can  be  solved  in  the  manner  explained  under  Case  I. 
The  graph  of  equation  (1)  is  too  complicated  to  be  given  here. 
(h)  Solve  the  equations 

,      xy^U.  (2) 

If  we  multiply  each  member  of  (2)  by  2  and  add  the  members  of  the 
resulting  equation  to  the  corresponding  members  of  (1),  we  get 

x2  -\-2xy  +  y'^  =  M.  (3) 

Hence  x  +  y  =  8,  (4) 

or  x-\-  y  =—^.  (5) 

We  shall  get  all  the  solutions  of  (1)  and  (2)  by  solving  (2)  and  (4) 
and  also  (2)  and  (5). 


120  COLLEGE  ALGEBRA 

EXERCISES 

Solve  each  of  the  following  pairs  of  equations.  When 
neither  of  the  equations  is  of  third  degree,  check  your  results 
by  means  of  graphs.  When  one  of  the  equations  is  of  the 
third  degree,  check  your  results  by  direct  substitution  in  both 
the  original  equations. 

1.  ic2  +  2/2  =  io,  .    6.  a^+16/  =  44,      11.  y?-f=2, 
0^2/ =  3.  0^2/4-1  =  0.  x-y^\. 

2.  x^-f  =  %  1.  «-^  + 2/^  =  64,  12.  16a?2  +  25/=9, 
x  —  y  =  2.                        X  -\-y  =  ^.  xy  =  l. 

3.  So?  +  21f=^l,         8.  x'-f  =  l,  13.  64ar^  +  ?/3  =  4, 
2x  +  3y  =  l.                 x-y=.l.  4,x-\-y  =  2. 

4.  a;2_^  1/2  =  20,  9.  x'  +  y^=16,  14.  x'-^y^^S, 
icy  =  S.                            ^y=  10.  xy  =  2. 

5.  4a^  +  9?/2  =  l,         10.  ^  +  f  =  %  15.  a'?/  +  4  =  0, 
xy  =  2.                           x  +  y  =  l.  a;^  +  9  2/^  =  25. 

PROBLEMS 

rv      Q «    1-   The  sum  of  the  squares  of  two  numbers  is  193  and  the 
iK      product  of  the  numbers  is  84.     What  are  the  numbers  ? 

2.  The  sum  of  the  squares  of  two  numbers  is  250  and  the 
sum  of  the  numbers  is  22.     What  are  the  numbers  ? 

3.  The  difference  of  the  cubes  of  two  numbers  is  819, 
and  the  difference  of  the  numbers  is  3.  What  are  the 
numbers  ? 

4.  It  is  known  that  the  sum  of  the  lengths  of  two  cubical 
bins  is  29  feet  and  that  their  combined  capacity  is  6119  cubic 
feet.     What  are  the  dimensions  of  each  bin? 

5.  The  area  of  a  right  triangle  is  30  square  inches  and  its 
hypotenuse  is  13  inches.     Find  the  lengths  of  the  legs. 


SYSTEMS  OF  EQUATIONS  IN  TWO  UNKNOWNS      121 

6.  The  diagonal  of  a  rectangular  field  containing  11^  acres 
is  80  rods.     What  are  the  dimensions  of  the  field  ? 

7.  Two  circles  with  their  centers  on  the  same  diameter  of 
a  third  circle  are  tangent  to  each  other  externally  and  to  the 
third  circle  internally,  as  in  the  figure. 
What  must  be  the  radii  of  the  two 
inner  circles  in  order  that  the  sum 
of  their  areas  be  one  half  the  area  of 
the  outer  circle,  the  radius  of  the 
latter  being  10  inches  ? 

8.  What  must  be  the  radii  of  the 
inner  circles  of  Problem  7  in  order 
that  the  sum  of  their  areas  be  three 
fourths  that  of  the  outer  circle  ?  How  will  the  sum  of  the 
circumferences  of  the  inner  circles  compare  with  the  circum- 
ference of  the  outer  circle  ? 

9.  What  is  the  least  possible  combined  area  of  two  inner 
circles  related  to  a  circle  of  radius  10  inches  in  the  way  de- 
scribed in  Problem  7  ? 

Hint.  —  Let  this  combined  area  be  wa,  and  then  find  the  least  value 
of  a  for  real  values  of  the  radii  of  the  inner  circles. 

10.  Two  spheres  with  their  centers  on  the  same  diameter 
of  a  third  sphere  are  tangent  externally  to  each  other  and  in- 
ternally to  the  third  sphere.  What  must  be  the  radii  of  the 
two  inner  spheres  in  order  that  the  sum  of  their  volumes  be 
three  fourths  the  volume  of  the  outer  sphere,  the  radius  of  the 
latter  being  10  inches  ? 

11.  What  is  the  least  possible  total  volume  of  two  inner 
spheres  related  to  a  sphere  of  radius  10  inches  in  the  way  de- 
scribed in  Problem  10  ? 

12.  What  is  the  least  possible  total  surface  of  the  inner 
spheres  of  Problem  11  ? 


122  COLLEGE  ALGEBRA 

13.  Being  given  that  the  sides  AB,  BC,  and  CA  of  the  tri- 
angle ABC  are  8  inches,  12  inches,  and  15  inches  respectively, 

find  the  altitude  BD. 

Let  BD=x  and  AD  =  y.     Then  DC=15-y, 
and  we  have  the  two  equations 

x^  +  y^  =  64, 
x'^  +  (15  _  y)2  ^  144. 

These  equations  do  not  come  under  any  of  the  cases  we  have  discussed. 
The  method  of  solution  is,  however,  obvious. 

When  one  altitude  is  known,  the  others  can  easily  be  found  from  the 
fact  that  the  product  of  any  side  of  a  triangle  and  the  altitude  upon  that 
side  equals  twice  the  area  of  the  triangle. 

14.  Find  the  three  altitudes  of  the  triangle  whose  sides  are 
20,  24,  and  30  respectively. 

15.  Find  one  altitude  and  the  area  of  the  triangle  whose 
sides  are  7  centimeters,  10  centimeters,  and  15  centimeters 
respectively. 


CHAPTER  IX 
PROGRESSIONS 

91.  Arithmetic  progression.  —  A  succession  of  numbers  so 
related  that  each  one  is  obtained  by  adding  a  fixed  amount  to 
the  preceding  one  is  called  an  arithmetic  progression. 

The  numbers  forming  the  progression  are  called  its  terms. 
The  fixed  amount  which  must  be  added  to  any  term  to  get  the 
following  one  is  called  the  common  difference. 

92.  If  «!  is  the  first  term  of  the  progression  and  d  the  com- 
mon difference,  the  progression  is 

Oi,  ai  +  dj  %  +  2  (Z,  ai  4-  3  c?,  ••  • 

The  succession  of  dots  after  a-\-Sd  means  "  and  so  forth." 

For  example,  3,  7,  11,  15,  19,  23,  27; 

and  5,  3,  1,  _  1,   -  3,  -  5,  -  7,  -  9,   -  11 

are  arithmetic  progressions.     In  the  former  the  common  difference  is  4 
and  in  the  latter  it  is  —  2. 

If  a^  denotes  the  wth  term  of  an  arithmetic  progression,  it  is 

clear  that  ,  /       *x  ,  /t\ 

a„  =  ai  +  (n-l)A  (I) 

This  formula  involves  the  four  numbers  aj,  d,  n,  and  a„ ;  and 
by  means  of  it  we  can  find  any  one  of  them  when  the  other 
three  are  given. 

93.  The  sum  of  n  consecutive  terms  of  an  arithmetic  pro- 
gression. —  If  we  denote  by  s„  the  sum  of  the  first  n  terms  of 
an  arithmetic  progression,  then 

Sn  =  ai+[ai  +  d]  +  [ai  +2c?]+  ... +[ai +(n  -  3)d] 

+  [ai  +(n-  2)d^  +  [ai  +(n-  1)<?]. 

The  symbol  formed  by  the  three  dots  is  to  be  read  "  and  so 
on  up  to." 

123 


124  COLLEGE  ALGEBRA 

If  we  write  the  right  member  of  this  equation  in  the  reverse  order,  we 
have 

Sn=lai  +  in-  1)(Z]  +[ai  +(w  -  2)d]  +  [ai  +  (n  -  3)d]  +  ••. 

Adding  the  corresponding  members  of  these  two  equations,  we  get 

2sn  =[2ai+(w-l)d]  +  [2ai  +  (w-2)d+d]4-[2ai-f(»i-3)d+2d]+  ••• 
+  [2ai+(w-3)d+2d]  +[2  ai  -\-(n  -  2)d  +  d^-{-[2  ai  -{-  (n  -  l)dj. 

Now  there  are  n  brackets  in  the  right  member  of  this  equation  and  each 
of  them  is  equal  to  2  ai  +  (w  —  l)d.     Hence 

2  Sn  =  n[2  ai -\- (n  -  l)d], 
and  sn=^[2ai+(n-l)d]. 

If  we  take  into  consideration  the  value  of  a^  as  given  by 

Formula  I,  we  see  that 

s.  =  l{<'^  +  a,),  (II) 

Formulae  I  and  II  are  of  great  importance  in  problems  in- 
volving arithmetic  progressions.  They  involve  live  numbers 
tti,  d,  n,  a„,  and  s„.  If  any  three  of  these  are  given,  the  other 
two  can  be  found  by  means  of  these  formulae. 

94.  Series.  —  An  algebraic  expression  of  the  form 

«l  +  0f2  +  «3+    •••    +«n 

is  called  a  series.     The  a's  are  called  the  terms  of  the  series. 

When  we  speak  of  the  sum  of  a  series  we  mean  the  sum  in- 
dicated by  the  series.  Thus,  Formula  II  gives  us  the  sum  of 
an  arithmetic  series. 

95.  Arithmetic  means.  —  The  terms  of  an  arithmetic  pro- 
gression between  the  first  one  and  the  last  one  are  called  arith- 
metic means. 

By  the  aid  of  Formula  I  any  number  of  arithmetic  means 
can  be  inserted  between  two  given  numbers ;  that  is,  an  arith- 


PROGRESSIONS  125 

metic  progression  can  be  formed  with  the  two  given  numbers 
as  the  first  term  and  the  last  term  respectively  and  with  the 
given  number  of  intermediate  terms. 

Example.  —  Insert  7  arithmetic  means  between  2  and  26. 
Here  the  first  term  of  the  progression  to  be  formed  is  2  and  the  ninth 
one  is  26.    Besides  these  two  terms  there  are  to  be  7  intermediate  terms. 

Hence,  ai  =  2,  «  =  9,  ag  =  26. 

Substituting  these  values  in  Formula  I,  we  get 

26  =  2+(9-l)d 
Hence,  cZ  =  3, 

and  the  required  means  are 

5,  8,  11,  14,  17,  20,  23. 

The  student  is  doubtless  already  familiar  with  the  problem 
of  inserting  one  arithmetic  mean  between  two  numbers,  since 
this  is  the  same  as  the  problem  of  finding  the  average  of  two 
numbers.     If  M  is  the  average  of  a^  and  a^,  then 

and  Oj,  M,  and  a^  form  an  arithmetic  progression.  For  this 
reason  M  is  called  the  arithmetic  mean  of  a^  and  a^. 

EXERCISES   AND   PROBLEMS 

Determine  which  of  the  four  following  sets  of  numbers  form 
arithmetic  progressions : 

1.  7,  10,  13,  15.  3.   20, 13,  6,  -1,-8. 

2.  -4,0,4,8,12.  4.    1,  -1,-2. 

5.  Solve  Formula  I  for  «!,  n,  and  d  in  turn. 

6.  What  is  the  last  term  of  an  arithmetic  progression  of  12 
terms  if  the  first  term  is  —  2  and  the  common  difference  is  2  ? 

7.  There  are  20  terms  in  an  arithmetic  progression  of  which 
the  last  one  is  —  45,  and  the  common  difference  is  4i.  What 
is  the  first  term  ? 


126  COLLEGE  ALGEBRA 

8.  What  is  the  common  difference  when  27  is  the  first  of 
twelve  terms  of  an  arithmetic  progression  and  — 13  is  the 
last? 

9.  How  many  terms  are  there  in  an  arithmetic  progression 
of  which  the  first  term  is  321,  the  last  term  0,  and  the  com- 
mon difference  —  3  ? 

10.  Insert  four  arithmetic  means  between  7  and  20. 

11.  What  is  the  arithmetic  mean  of  20  and  —  20? 

12.  Insert  twelve  arithmetic  means  between  14  and  — 15, 
and  find  the  sum  of  the  terms  of  the  resulting  progression. 

13.  Given  Ui  =  \,  Sig  =  102 ;  find  d  and  ai2- 

14.  If  a  clock  strikes  the  hours,  how  many  strokes  a  day 
does  it  make  ? 

15.  Find  the  sum  of  the  first  fifty  odd  numbers. 

16.  Prove  that  the  sum  of  the  first  n  odd  numbers  is  n^. 

17.  A  freely  falling  body  falls  approximately  16  feet  the 
first  second  and  in  each  second  thereafter  32  feet  more  than  in 
the  preceding  second.  A  stone  dropped  from  the  top  of  a 
tower  strikes  the  ground  in  4  seconds.  How  high  is  the  tower 
and  how  far  did  the  stone  fall  the  last  second  ? 

18.  Find  the  sum  of  all  the  integers  between  100  and  200 
that  are  divisible  by  3. 

19.  A  ball  rolling  down  an  incline  of  30°  goes  8  feet  the  first 
second  and  in  each  second  thereafter  16  feet  more  than  in  the 
preceding  second.     How  far  will  it  roll  in  9  seconds  ? 

20.  Show  that  the  distances  of  the  following  points  from 
the  origin  are  in  arithmetic  progression : 

(1,  li),  (2,  2|),  (3,  4),  (4,  6i),  (5,  6f),  and  (6,  8). 

Plot  these  points  and  see  if  they  lie  on  a  straight  line. 


PROGRESSIONS  127 

96.  Geometric  progression.  —  A  succession  of  numbers  so 
related  that  the  ratio  of  each  one  to  the  preceding  one  is 
a  fixed  number  is  called  a  geometric  progression. 

Thus  the  numbers  3,  6,  12,  24,  48  and  96  form  a  geometric  progression. 

The  numbers  forming  a  progression  are  called  its  terms. 
The  ratio  of  any  term  to  the  preceding  one  is  called  the  com- 
mon ratio  of  the  progression,  and  is  usually  represented  by 
the  letter  r.  The  first  term,  the  n\h  term,  the  number  of 
terms,  and  the  sum  of  the  terms  are  represented  by  the  same 
letters  as  the  corresponding  numbers  in  an  arithmetic  pro- 
gression.    We  shall  assume  throughout  that  r4^1. 

97.  The  terms  of  a  geometric  progression  can  be  repre- 
sented in  the  following  way : 

«i,  cur,  a^r^,  a^i^,  —. 

From  this  it  is  obvious  that  the  nth.  term  of  the  progression 
is  ar"~^    That  is,  in  a  geometric  progression  of  n  terms 

a,=  ar-\  (I) 

This  formula  involves  the  four  numbers,  a^  a„,  n,  and  r; 
and  by  means  of  it  we  can  find  any  one  of  them  when  the 
other  three  are  given. 


^^dS.   The  sum  of  n  consecutive  terms  of  a  geometric  pro- 
-^gression. — 

«n  =  «i  +  «i^  +  «i^'^  +  •  •  •  +  aiV"-^  +  a^r^'-K 

Multiplying  each  member  of  this  equation  by  r,  we  get 

sj-  =  a^r  4-  a^r^  +  a^r^  +  •  •  •  +  ai^^-^  +  air". 
Hence,  by  subtraction. 

All  the  terms,  except  two,  in  the  right  members  of  these 
equations  are  destroyed  by  the  subtraction. 


128  COLLEGE  ALGEBRA 

Solving  the  last  equation  for  s„,  we  get 

^»=^i^-  (11) 

The  student  can  verify  that  Formula  II  is  correct  by  dividing  the 
numerator  of  the  right  member  by  the  denominator. 

If  we  take  Formula  I  into  consideration,  we  can  write 
Formula  II  in  the  following  form  : 

*„=^P^-  (m) 

1  —  r 

By  means  of  Formulae  I  and  II  or  I  and  III  we  can  find  any 
two  of  the  five  numbers  «!,  a„,  n,  r,  and  s„  when  the  other  three 
are  given. 

99.  Geometric  means.  —  The  terms  of  a  geometric  progres- 
sion between  the  first  one  and  the  last  one  are  called  geometric 
means. 

In  a  geometric  progression  of  three  terms  the  middle  term  is 
called  a  geometric  mean  of  the  other  two. 

By  the  aid  of  Formula  I  any  number  of  geometric  means  can 
be  inserted  between  two  given  numbers. 

Example.  — Insert  four  geometric  means  between  5  and  160. 

In  the  formula 

an  =  air"--^ 

«!  =  5,  w  =  6,  and  a&  =  160  since  there  are  to  be  four  terms  besides  the 
first  one  and  the  last  one.     Hence 

5  r6  =  160, 
r5  =z  32, 
r  =  2, 
and  the  required  geometric  means  are  10,  20,  40,  and  80. 

In  discussing  arithmetic  progressions  we  found  only  one 
way  to  insert  the  required  number  of  arithmetic  means  be- 


PROGRESSIONS  129 

tween  two  given  numbers.  But  the  required  number  of  geo- 
metric means  can  be  inserted  between  two  given  numbers  in 
more  than  one  way. 

Suppose,  for  example,  we  wish  to  insert  three  geometric  means  between 
6  and  1536.     Here  ai  =  6,  n  =  o,  and  a^  =  1636.     Hence 

6r*  =  1536, 
r*  =  256, 

r  =  ±4. 

The  required  means  are  therefore  either 
24,  96,  and  384  ; 
or  -  24,  96,  and  -  384. 

In  this  example  the  common  ratio  can  be  any  number  whose  fourth 
power  is  266.  Now  it  will  be  shown  in  §  138  that  there  are  four  such 
numbers.  However,  only  two  of  them  are  real,  and  we  are  considering 
here  only  real  numbers.  This  explains  why  in  the  first  example  we  gave 
only  one  set  of  geometric  means.  As  a  matter  of  fact,  there  are  five  such 
sets,  since  there  are  five  numbers  whose  fifth  power  is  32. 

The  problem  of  inserting  one  geometric  mean  between  two 
numbers  is  the  same  as  that  of  finding  a  mean  proportional 
between  these  numbers. 

Two  real  numbers  of  opposite  signs  have  no  real  geometric 
mean,  since  the  square  of  such  a  mean  would  necessarily  be 
negative. 

EXERCISES  AND   PROBLEMS 

Determine  which  of  the  four  following  sets  of  numbers  form 
geometric  progressions  : 

— 1.    -  2,  4,  -  8, 16.  ^ 3.    -  2,  -  4,  -  8,  -  16. 

2.    5,7,10,13.  4.    1,4,16,32. 

5.    Solve  Formula  I  for  a^  and  r  in  turn. 

"  6.  Find  the  tenth  term  of  a  geometric  progression  whose 
first  three  terms  are  4,  |,  and  g\  respectively. 


<l 


130  COLLEGE  ALGEBRA 

7.  The  first  and  fourth  terms  of  a  geometric  progression 
are  2  and  54  respectively.  Find  the  common  ratio  and  the 
second  term. 

""^  8.    What  is  the  fifth  term  of  a  geometric  progression  whose 
first  two  terms  are  a  and  h  respectively  ? 

9.  The  first  term  of  a  geometric  progression  is  7  and  the 
common  ratio  is  3.     What  is  the  sum  of  the  first  eight  terms  ? 

10.  Insert  three  geometric  means  between  6  and  486. 

11.  Which  is  the  greater,  the  arithmetic  mean  of  4  and  16, 
or  the  positive  geometric  mean  of  these  numbers  ? 

12.  Write  both  the  arithmetic  mean  and  the  positive  geo- 
metric mean  of  the  positive  numbers  a  and  h.  Then  select  four 
sets  of  values  for  a  and  h  and  determine  which  of  these  means 
is  the  greater  for  these  values  of  a  and  h. 

13.  What  will  $100  amount  to  in  five  years  at  3%,  com- 
pounded annually  ?  /  \\  r.   .      \  b  o\\'0^jJ 

14.  What  will  p  dollars  amount  to  in  n  years  at  r  per  cent, 
compounded  annually  ?  compounded  quarterly  ? 

15.  Solve  Formula  III  for  each  letter  in  terms  of  the  others. 

16.  Show  that  the  distances  from  the  origin  of  the  points 
(I,  1),  (f,  2),  P/,  4),  {^S  8),  and  (-V,  16)  form  a  geometric 
progression  and  find  their  sum. 

17.  Each  stroke  of  an  air  pump  exhausts  one  fifteenth  of 
the  air  in  the  receiver.  How  much  of  the  air  originally  in  the 
receiver  is  removed  in  ten  strokes  ? 

18.  Given  aj  =  3,  r  =  2  ;  find  a^  and  s^. 

19.  The  perpendicular  distance  from  the  center  of  a  circle  to 
a  side  of  an  inscribed  equilateral  triangle  is  one  half  the  radius 
of  the  circle.  An  equilateral  triangle  is  inscribed  in  a  circle 
whose  radius  is  6  inches.  Then  a  circle  is  inscribed  in  the 
triangle  and  an  equilateral  triangle  in  this  circle  and  so  on  until 
there  are  five  circles  in  all.  What  is  the  sum  of  their  circum- 
ferences ?    What  is  the  sum  of  the  perimeters  of  the  triangles  ? 


PROGRESSIONS  131 

20.  The  second  of  a  series  of  squares  has  its  vertices  at  the 
mid-points  of  the  sides  of  the  first  one,  and,  in  general,  each 
square  of  the  series  has  its  vertices  at  the  mid-points  of  the 
sides  of  the  preceding  one.  What  are  the  area  and  perimeter 
of  the  tenth  square,  if  each  side  of  the  first  square  is  2  inches  ? 

100.  Limit  of  certain  geometric  series.  —  Consider  the  geo- 
metric progression  whose  first  term  is  1  and  whose  common 
ratio  is  ^.     The  sum  of  the  first  n  terms  is  given  by  the  formula 

It  will  be  observed  that  this  sum  is  always  less  than  2,  no 
matter  how  many  terms  of  the  progression  we  take  ;  but  that 
it  differs  from  2  by  an  amount  that  decreases  as  the  number 
of  terms  taken  increases.  By  adding  together  enough  terms 
of  this  series  we  can  get  a  sum  that  differs  from  2  by  as  small 
an  amount  as  we  please,  and  the  difference  will  be  still  less  if 
we  take  more  terms.  Thus  the  sum  of  ten  terms  lacks  -^^^  ^^ 
being  equal  to  2,  and  the  sum  of  twenty  terms  lacks  -^jtt^  ^^ 
being  equal  to  2. 

We  express  these  facts  concisely  by  saying  that  2  is  the 
limit  of  the  sum  of  7i  terms  of  this  progression  as  n  is  increased 
without  limit. 

In  general,  we  have 

a(l  —  r")         a  a       „ 

1  —  r         1  —  ?•     1  —  r 

As  we  give  to  n  larger  and  larger  values,  the  fraction 


1-r 

does  not  change,  while  if  the  absolute  value  of  r  is  less  than  1, 

•  r"  gets  nearer  and  nearer  to  zero,  and  we  can  make  it 

1  —  r 

as  near  to  zero  as  we  wish  by  making  n  large  enough.*    This  is 

*Not  all  variables  that  are  getting  nearer  and  nearer  to  zero  can  get  as 
near  to  zero  as  we  wish.  A  variable  might,  for  example,  be  constantly  getting 
smaller  and  yet  never  get  as  small  as  one. 


132  COLLEGE   ALGEBRA 

due  to  the  fact  that  by  taking  n  large  enough  we  can  make  r" 
as  near  to  zero  as  we  wish.     Hence  «„  can  be  made  to  differ 

from by  as  little  as  we  please  by  making  n  large  enough. 

1  —  r 

We  express  this  by  saying  that is  the  limit  of  s^  as  n  is 

1  —  r 

increased  without  limit,  or  that  ^  ^     is  the  limit  of  the  sum  of 

l  —  r 

the  geometric  series  whose  common  ratio  is  r,  as  the  number 

of  terms  is  increased  without  limit.     (See  §  202.) 

When  the  absolute  value  of  r  is  greater  than  1,  the  fraction 

•  r"  does  not  get  nearer  to  zero  as  n  increases.     Hence 

l  —  r 

the  preceding  argument  does  not  apply.     Such  infinite  series 

are  not  considered  here.     (See  §  203.) 

The  repeating  decimal       .666  ••• 

can  be  written  thus  .6  +  .06-f- .006+  .... 

It  is  therefore  an  infinite  geometric  series  whose  first  term 
is  .6  and  whose  common  ratio  is  .1.     Hence 

•^  -^     (.!)«. 


.1      1  -  .1 


6  2 

This   approaches   the   limit    — ^ — -  or  -   when  n   increases 

without  limit.  "" ' 


EXERCISES 

Find  the  limit  of  the  following  geometric  series  as  the  num- 
ber of  terms  increases  without  limit. 

1.  l  +  i  +  i+    ....  5.  3__9^+2|_    .... 

2.  1-i  +  i-  .-..  -  6.  4  +  I  +  V-+  -. 

3.  5-hJ/  +  f|+  ....                  7.  2  +  1  +  1+  .-. 
_    4.    6-4  +  1-  ....                 -8.  2-l  +  i-  .... 


PROGRESSIONS  133 

Find  the  limiting  value  of  each  of  the  following  repeating 
decimals: 

9.  .333  ....  15.  3.544242  .... 

10.  .07777777  ....  16.  .0065334334  .... 

11.  1.66666  ....  17.  1.21212  .... 

12.  1.111  «...  — *18.  32.34125125  .... 

13.  20.202020  ....  19.  .00041616  .... 

14.  2.555  ....  ~   20.  5.2330404  .... 

^AArJr^^^'  H^^^o^^c  progression.  —  A  succession  of  numbers  so 
related  that  their  reciprocals  form  an  arithmetic  progression 
is  called  a  harmonic  progression. 

Thus  the  numbers 

1»     h     h     h     h     etc. 
form  a  harmonic  progression,  and  it  is  from  this  particular  progression 
that  the  name  "harmonic"  is  derived.     If  a  set  of  stretched  wires  or 
strings  whose  lengths  are  proportional  to  the  terms  of  this  progression 
are  set  vibrating,  the  sound  produced  is  harmonious. 

We  shall  have  no  occasion  in  what  follows  to  use  harmonic 
progressions. 

Ex.  What  can  you  say  of  a  succession  of  numbers  so  re- 
lated that  their  reciprocals  form  a  geometric  progression? 


CHAPTER   X 
PERMUTATIONS  AND   COMBINATIONS 

102.  Permutations. —  In  certain  problems  it  is  necessary  to 
find  out  in  how  many  orders  a  given  number  of  things  can  be 
arranged.  This  information  is  obtained  by  counting.  But  it 
is  not  necessary  actually  to  make  this  count  in  every  problem 
that  presents  itself.  The  result  obtained  in  a  typical  problem 
can  be  embodied  in  a  formula,  and  then,  in  order  to  solve  any 
problem  of  this  type,  we  have  only  to  substitute  the  proper 
numbers  in  this  formiTla. 

Each  arrangement  of  a  number  of  things  in  a  d^nite  order  is  jr  ^ 
called  a  permutation  of  these  things.  V 

Suppose  we  have  three  things  represented  by  the  letters 
a,  bf  c  respectively.  They  can  be  arranged  in  the  six  following 
orders:  a    b    c  b    a    c  cab 

a    c    b  b     c    a  c    b     a 

That  is,  six  different  permutations  of  three  things  are  possible. 

Sometimes  when  we  are  considering  n  things  and  r  is  some 
positive  integer  less  than  n,  we  wish  to  find  the  total  number 
of  possible  permutations  of  these  n  things  when  in  each  per- 
mutation we  use  only  r  things,  or,  as  we  shall  say,  the  number 
of  permutations  of  n  things  taken  r  at  a  time.  Thus,  all  the 
possible  permutations  of  four  things  taken  three  at  a  time  are 
given  in  the  following  table : 


a 

b 

c 

a 

b 

d 

a 

c 

d 

b 

c 

d 

a 

c 

b 

a 

d 

b 

a 

d 

c 

b 

d 

c 

b 

a 

c 

b 

a 

d 

c 

a 

d 

c 

b 

d 

b 

c 

a 

b 

d 

a 

c 

d 

a 

c 

d 

b 

c 

a 

b 

d 

b 

a 

d 

c 

a 

d 

c 

b 

c 

b 

a 

d 

a 

b 

d 

a 

c 

d 

b 

c 

It  is  easy  to  see  by  counting  that  there  are  24  of  these. 

134 


PERMUTATIONS  AND   COMBEsFATIONS  135 

In  order  to  derive  a  general  formula  for  the  number  of  per- 
mutations of  n  tilings  taken  r  at  a  time,  r  being  a  positive 
integer  less  than,  or  equal  to,  n,  we  shall  consider  first  a  pre- 
liminary tlieorem. 

103.  Illustrative  example.  —  If  there  are  two  roads  from  A  to 
B  and  three  from  B  to  C,  by  how  many  routes  can  one  go  from 
A  through  B  to  C  ? 

Either  of  the  roads  from  A  to  B  can  be  followed  by  any  one 
of  the  three  from  B  to  C.  Hence  there  are  3  •  2,  or  6,  routes  by 
which  one  can  go  from  A  through  B  to  C. 

This  suggests  the  following  general 

Theorem. — If  one  act  can  he  performed  in  p  ivays;  and  if 
after  this  has  been  performed  in  one  of  these  ways,  a  second  act 
can  be  performed  in  q  ways,  then  the  two  can  be  performed  in  this 
order  in  pq  ways. 

The  truth  of  this  theorem  follows  when  we  consider  that 
any  one  of  the  p  ways  of  performing  the  first  act  can  be  fol- 
lowed by  any  one  of  the  q  ways  of  performing  the  second  act. 

The  following  is  a  generalization  of  this  theorem. 

If  one  act  can  be  performed  in  p  ways,  then  a  second  in  q  ways, 
then  a  third  in  r  ways,  and  so  on,  they  can  all  be  performed  in 
this  order  in  pqr  •  •  •  ways. 

Ex.  Prove  this  generalization  for  the  case  in  which  there  are  three 
acts  to  be  performed. 

104.  The  ntmiber  of  permutations  of  n  things  taken  r  at  a 
time.  —  We  are  now  ready  to  determine  the  number  of  permu- 
tations of  2j^ings  taken  r  atj,  time.  We  shall  represent  this 
number  by  the  symbol  „P, ,  and  the  things  to  be  arranged  by 
the  letters  a^,  ofg)  •••  ««• 

For  the  first  place  in  the  arrangement  we  have  the  choice  of 
any  one  of  the  n  things.  When  this  place  has  been  filled  we 
have  the  choice  of  any  one  of  the  remaining  n  —  1  things  for 
the  second  place.     When  these  two  places  have  been  filled,  we 


136  COLLEGE  ALGEBRA 

have  the  choice  of  any  one  of  the  remaining  n  —  2  things  for 
the  third  place.  When  k  places  have  been  filled  we  have  the 
choice  of  any  one  of  the  remaining  n  —  k  things  for  the 
(k-\-l)th  place.  When  all  but  the  last,  or  rth,  place  has  been 
filled,  we  have  the  choice  of  any  one  of  the  remaining  n—  (r— 1), 
or  n  —  r  +  lj  things  for  this  place.  Hence,  by  virtue  of  the 
generalization  of  the  preliminary  theorem, 

„P,  =  n{n  -  l)(n  -  2)  ...  (n  -  r  +  1).  (1) 

The  student  should  observe  that  the  first  factor  in  this 
formula  is  the  total  number  of  things  considered  and  that  the 
number  of  factors  is  the  number  of  things  taken  in  each 
arrangement. 

Factorial  n.  —  The  product  of  all  the  integers  from  1  to  n  in- 
clusive is  called  factorial  n  and  is  represented  by  the  symbol  n  !. 
If  in  each  arrangement  we  take  all  the  things ;  that  is,  if 
•    r  =  71,  we  get  the  formula 

„P^  =  w(7i  -  1)  ...  (n  -  n  +  2)(n  -  n  +  1) 

=  7i(7l-l)...2.1 

=  n!  (2) 

Formula  (2)  is  equivalent  to  the  statement  that  the  number 
of  permutations  of  n  things  taken  n  at  a  time  is  factorial  n. 

EXERCISES 

1.   In  how  many  ways  can  I  make  the  trip  across  a  lake  and 
)       back  if  I  can  go  over  in  any  one  of  five  rowboats  and  return 

in  any  one  of  four  launches  ? 
^'        2.  How  many  whole  numbers  of  four  unlike  digits  each  can 

bejormed  from  the  digits  1,  2,  3,  4,  5,  6,  7  ? 
^    '■      3.   How  many  signals  of  three  flags  each,  in  vertical  order, 
^  ican  be  made  from  eight  flags  all  different ? 

4.   How  many  whole  numbers  less  than  500  and  containing 
</     no  repetition  of  digits  can  be  formed  from  the  digits  1,  2,  3,  4, 

and  5? 


7- 


PERMUTATIONS  AND  COMBINATIONS  137 

5.  In  how  many  ways  can  eight  books  with  blue  bindings 
and  four  with  olive-green  bindings  be  so  arranged  on  a  shelf 
that  those  with  the  same  binding  shall  be  together  ? 

6.  For  a  certain  value  of  n,  ^P^=9^P^.     Find  this  value.   /  '^ 

7.  For  a  certain  value  of  r,  ^oPr  =  ^nPr-     Find  this  value.   ^ 

8.  For  a  certain  value  of  r,  ^qP^  =  SgP^.     Find  this  value.  6f 

9.  In  how  many  ways  can  crops  of  corn,  wheat,  oats,  and 
barley  be  raised  in  four  fields,  each  field  being  used  for  a  single 
crop  ?  ^2    ^ 

10.  In  how  many  ways  can  ten  keys  be  arranged  on  a  ring  ? 

Suggestion.  —  The  position  of  one  key  is  immaterial.  The  only  thing 
that  needs  to  be  considered  is  the  positions  of  the  other  nine  relative  to 
this  one.  3  (>  2^^  ^  0  -  ^\\ 

11.  In  how  many  different  relative  positions  can  a  party  of 

six  people  seat  themselves  at  a  round  table  ?        /  '2-  S 

12.  Given  ^P^  =  12  •  ^P, ;  solve  for  n.      7 

13.  A  railway  signal  has  three  arms  and  each  arm  can  be 
put  into  two  positions  besides  its  position  of  rest.  How  many 
signals  can  be  given  by  it?  Every  position  of  the  arms, 
except  that  in  which  they  are  all  at  rest,  forms  a  signal.    1^ 

14.  How  many  kinds  of  single  trip  local  passenger  tickets, 
good  either  way,  will  a  railway  company  need  for  use  on  a 
division  that  has  ten  stations  ?  t/  C'  .  C  <2. 

15.  In  how  many  different  relative  positions  can  a  party  of 
five  ladies  and  five  gentlemen  be  seated  at  a  round  table,  the 
ladies  and  gentlemen  being  seated  alternately  ?     '2-g's' '-' 

105.  Distinguishable  permutations  of  n  things  not  all  differ- 
ent. —  The  number  of  arrangements  of  the  four  letters  of  the 
word  city  is  4!,  or  24;  but  the  number  of  distinguishable 
arrangements  of  the  six  letters  of  the  word  Canada  is  not  6 ! . 

This  is  due  to  the  fact  that  these  six  letters  are  not  all  dis- 
tinct.    Any  rearrangement  of  the  three  a's  does  not  make  a 


138  COLLEGE  ALGEBRA 

distinguishable  difference  in  any  of  the  arrangements.  Hence 
for  every  arrangement  of  these  letters  there  would  be  3 !,  or  6, 
arrangements  if  the  letters  were  all  distinct.     The  number  of 

A  t 

distinguishable  arrangements  is  then  — ^,  or  120. 

o ! 

In  general,  for  every  arrangement  of  n  things  p  of  which  are 
alike,  there  would  be  p !  arrangements  if  these  n  things  were 
all  distinct.     Hence, 

Theorem.  —  If  of  n  things  p  are  alike,  the  number  of  per- 

n ' 
mutations  of  these  things  taken  all  together  is  —^ . 

p\ 

Corollary.  —  If  of  n  things  p  are  of  one  kind,  q  of  another,  r  of 
another,  and  so  on,  the  number  of  permutations  of  these  things 

taken  all  together  is  '- . 

p\  '  q\  '  r\  '•' 

The  proof  is  left  to  the  student. 

EXERCISES 

1.  How  many  permutations  are  there  of  the  letters  of  the 
word  permutation  ?     f  'f  /  ^  ^)  ^  0'^-' 

2.  How  many  signals  can  be  made  by  displaying  twelve 
flags  one  above  the  other,  of  which  four  are  white,  three  blue, 
one  red,  and  the  rest  black  ?       /  '3  1'^  .- 

3.  How  many  signals  can  be  made  by  displaying  the  flags 

of  Ex.  2  if  the  top  flag  must  be  red  and  the  bottom  one  blue  ?  ^  '  ^ 

4.  In  how  many  ways  can  a  row  of  five  white  balls,  four  red 
ones,  and  two  black  ones  be  arranged  ?     Q^  ^3  0 

5.  Four  dimes,  three  quarters,  and  three  half-dollars  are  to 

be  distributed  among  ten  boys  in  such  a  way  that  each  boy  ^  ^ 
shall  receive  one  coin.     In  how  many  ways  can  this  be  done  ? 

106.  Combinations.  —  If  in  any  set  of  things  we  are  interested 
only  in  the  things  themselves  and  do  not  consider  the  order  in 
which  they  g/my  be  arranged,  we  speak  of  the  set  as  a  combination. 


PERMUTATIONS  AND  COMBINATIONS  139 

There  is  obviously  only  one  combination  of  n  things  taken  n  at 
a  time,  but  there  is  more  than  one  combination  of  n  things  taken 
r  at  a  time,  if  r  is  less  than  n.  In  certain  problems  it  is  neces- 
sary to  determine  this  number.  We  shall  accordingly  derive  a 
general  formula  for  it,  and  shall  represent  it  by  the  symbol  „C,. 

107.  Number  of  combinations.  —  Consider  first  the  combina- 
tions of  6  things  taken  4  at  a  time.  We  can  determine  in  the 
following  way  just  how  many  of  these  there  are. 

The  4  things  of  any  one  of  these  combinations  can  be  ar- 
ranged in  4 !  different  orders  and  these  different  arrangements 
give  rise  to  4 !  distinct  permutations  of  the  6  things  taken  4  at 
a  time.  Moreover  every  permutation  of  the  6  things  4  at  a 
time  is  related  to  some  one  of  these  combinations  in  this  way. 

6^4  •  ^  i  —  6^4) 

and  therefore  e^*  = 


4! 

In  general,  since  there  are  r!  permutations  of  r  things  taken 
r  at  a  time,  it  follows  that  for  every  combination  of  n  things 
taken  r  at  a  time  there  are  r !  distinct  permutations  of  these  n 
things  taken  r  at  a  time.  Moreover  every  permutation  of  these 
n  things  r  at  a  time  is  related  to  some  one  of  these  combina- 
tions in  this  way.     Hence, 

and  therefore   ^c,  =  '>{n-\){n-Z)...{n  -  r+\)  ^  ^ 

If  we  multiply  both  numerator  and  denominator  of  this  ex- 
pression for  ^C^  by  (ii  —  r)  !  we  get 

C  =  ^  (:n  —  1)  (n  —  2)  • . .  (n  —  r  4- 1)  {n  —  r)  !  ^  n\  ,^. 

"    '  r!  '{n-r)l  rl  '(n-r)l'  ^  ^ 

108.  Whenever  we  make  a  combination  of  r  things  from  n 
things,  there  is  left  a  combination  of  (n  —  r)  th:       ,  and  two 


140  COLLEGE  ALGEBRA 

different  combinations  of  r  things  from  the  same  n  things  leave 
different  combinations  of  {n  —  r)  things.     Hence, 

The  number  of  combinations  of  n  things  taken  r  at  a  time  is 
equal  to  the  iiumber  of  combi7iations  of  n  things  taken  n  —  r  at  a 
time.     That  is,  ^O^  =  „(7„_^. 

Prove  this  last  statement  directly  from  Formula  (4). 

PROBLEMS 

1.  How  many  straight  lines  can  be  drawn  through  pairs  of 
points  selected  from  nine  points  no  three  of  which  are  in  a 
straight  line?  T,i>Q,i -^Ij<^i} 

¥ic%(kf^      2.    If  the  lines  of  Ex.  1  are  produced  indefinitely  and  no  7t^ 
'  0  ■'     two  of  them  are  parallel,  how  many  triangles  will  be  formed  ?-p^ 

3.    If  no  four  of  a  set  of  fourteen  points  lie  in  one  plane, 
^Q  \    how  many  tetrahedra  are  there  whose  vertices  are  in  this  set ? 

t  4.   How  many  planes  are  determined  by  the  points  of  Ex.  3  ? 

A  plane  is  determined  by  three  points  not  in  a  straight  line. 

5.  At  a  meeting  of  ten  men  each  shakes  hands  with  all  the 
others.     How  many  handshakes  are  there  ? 

G.    How  many  committees  each  consisting  of  five  Republi- 
1 7^8'/53,ns  and  four  Democrats  can  be  chosen  from  twenty  Republi- 
cans and  sixteen  Democrats  ? 

d^         7.    How  many  parallelograms  are  formed  when  a  set  of  nine 
'  parallel  lines  is  met  by  another  set  of  eight  parallel  lines  ? 

8.  In  how  many  ways  can  seven  ladies  and  four  gentlemen 
I*"  •»      arrange  a  game  of  tennis,  each  side  to  consist  of  one  lady  and 

one  gentleman  ? 

9.  In  how  many  ways  can  a  baseball  nine  be  formed  from 
sixteen  players,  of  whom  nine  can  play  only  in  the  infield  and 

jK      seven  only  in  the  outfield?     Each  of  the  infield  players  can 
iVr^     play  any  one  of  the  six  infield  positions,  and  each  of  the  out- 
field players  can  play  any  one  of  the  three  outfield  positions. 


//- 


ffc 


ui 


PERMUTATIONS  AND  COMBINATIONS  141 

10.  In  how  many  ways  can  a  committee  of  five  men  be  se-  /  ; 
lected  from  a  group  of  twelve  men  ? 

11.  How  many  sub-committees,  each  containing  five  mem- 
bers, of  whom  at  least  three  shall  be  Democrats,  can  be  formed 
from  a  committee  of  eight  Democrats  and  five  Republicans  ? 

.  12.    Given  „P,  =  20,  „0,  =  10;  find  n  and  r.        ^t  ^  s'  r  ~ 

13.  How  many  groups  of  three  letters  each  can  be  formed 
from  the  letters  of  the  alphabet  ? 

14.  In  how  many  ways  can  a  picket  of  five  men  be  formed 
from  a  company  of  100  soldiers  ?         i  ^^l^%  '/  S    1^  O 

15.  How  many  combinations  each  consisting  of  four  red 
balls,  two  white  ones,  and  five  black  ones  can  be  formed  from 
seven  red  balls,  six  white  ones,  and  nine  black  ones  ? 

16.  In  how  many  ways  can  a  committee  of  four  lawyers, 
three  merchants,  and  one  physician  be  formed  from  the  stock- 
holders of  a  corporation  consisting  of  ten  merchants,  eight 
lawyers,  and  five  physicians.?  /^.  l^-  2^  o  0  0  . 

17.  How  manylairnngementsjof  four  consonants  and  three 
vowels  can  be  made  from  eight  consonants  and  five  vowels  ? 

109.   The  binomial  theorem  for  positive  integral  powers. —       / 

Consider  the  product      ,     ,  ,.,     ,  ,.,     ,  ,v 

of  3  equal  binomial  factors.  The  product  of  the  first  terms  of 
all  the  factors,  namely  o?,  is  a  term  of  the  product.  So  also  is 
the  product  of  the  first  terms  of  any  two  of  the  factors  and  the 
second  term  of  the  remaining  factor.  This  is  a^h.  Moreover 
there  are  three  such  terms  in  the  product  since  the  factor  whose 
second  term  is  used  may  be  any  one  of  the  three  factors. 
Hence  when  like  terms  are  combined  the  product  will  contain 
the  term  3  a^h.  It  is  evident  from  the  way  this  term  was  ob- 
tained that  in  it  the  coefficient  of  a}h  is  equal  to  the  number  of 
ways  in  which  the  term,  h  can  be  selected  from  the  three  given 
factors,  and  this  number  is  fix-     The  product  of  the  first  term 


142  COLLEGE  ALGEBRA 

of  any  factor  and  the  second  terms  of  the  other  factors,  namely 
aJy^j  is  also  in  the  product ;  and  there  are  as  many  such  terms 
as  there  are  ways  of  selecting  the  term  b  from  two  of  tlie  three 
factors.  Hence,  when  like  terms  are  combined,  the  coefficient 
of  a¥  is  3C2.  Finally  the  product  contains  the  product  of  the 
second  terms  of  all  the  factors.     Hence, 

(a  +  hf  =  a?  +  sC^a^b  +  Aa¥  +  b\ 

Consider  now  the  product 

(a  +  b){a  +  b)  ...(a +  6) 

of  n  equal  binomial  factors.  The  product  of  the  first  terms  of 
all  the  factors,  namely,  a%  is  a  term  of  the  product.  So  also  is 
the  product  of  the  first  terms  of  any  n  —  1  of  the  factors  and 
the  second  term  of  the  remaining  factor.  This  is  a""^6.  More- 
over there  are  n  such  terms  in  the  product  since  the  factor 
whose  second  term  is  used  may  be  any  one  of  the  n  factors. 
Hence  when  like  terms  are  combined  the  product  will  contain 
the  term  na'^'^b.  It  is  evident  from  the  way  this  term  was  ob- 
tained that  in  it  the  coefficient  of  a'*~^6  is  equal  to  the  number 
of  ways  in  which  the  term  b  can  be  selected  from  the  n  given 
factors,  and  this  number  is  „(7i.  The  product  of  the  first  terms 
of  any  n  —  2oi  the  factors  and  the  second  terms  of  the  remain- 
ing factors,  namely,  a^~^b^,  is  also  in  the  product ;  and  there  are 
as  many  such  terms  as  there  are  ways  of  selecting  the  term  b 
from  two  of  the  n  factors.  Hence  when  like  terms  are  com- 
bined the  coefficient  of  a'*"^^^  is  „(72.  In  a  similar  way  it  can  be 
seen  that  in  the  product  the  coefficient  of  a"~''6'",  where  r  is  any 
integer  from  1  to  w  inclusive,  is  „(7^.  Hence  for  any  positive 
integral  value  of  n, 

The  statement  of  this  relation  is  called  the  binomial  theorem 
for  positive  integral  powers.  The  right  member  of  this  iden- 
tity is  called  the  expansion  of  (a  +  by. 


PERMUTATIONS  AND  COMBINATIONS  143 

If  we  take  into  account  the  values  of  ^(7,  for  different  values 
of  r,  as  given  in  §  107,  we  can  write  the  binomial  theorem  as 
follows : 

(a  +  &)^  =  fl^  +  na^-^b  +  "(^  ~  ^)  0^-2^2  4.  ... 
2  ! 

110.  The  term  whose  coefficient  is  „(7^  is  the  (r-f  l)th  term 
from  the  left  and  the  (n  + 1  —  r)th  term  from  the  right.  It  is 
called  the  general  term  since  any  term  except  the  first  one  can 
be  obtained  from  it  by  substituting  in  it  the  proper  value  of  r. 

The  term  whose  coefficient  is  „(7^  will  be  as  far  from  the  right 
end  of  the  expansion  as  „(7^  is  from  the  left  end  if 

r4-l=7i4-l  —  s; 
that  is,  if  s  =  n  —  r. 

Hence  the  terms  whose  coefficients  are  ^C^  and  „(7„._^  respect- 
ively are  equidistant  from  the  ends.  But  ^C^  =  n^n-r  (§  108). 
This  proves  that  the  coefficients  of  any  two  terms  equidistant  from 
the  ends  of  the  expansion  of  (a-\-by  are  equal. 

111.  Greatest  coefficient.  —  Since 

^  _7i(n  —  l)(n  —  2)  '•'  (n  —  r-^1) 
r  I 

and  ^     _n(n-l)(n-2)  .-»  (n-r-^2) 

^^-^-  F=^iy^  ' 

it  follows  that  ,  ^ 

^  __n-r-\-l  ^ 

r 

Hence  ^C^  is  greater  than   the  preceding  coefficient  when 

n-r-\-l^^     that  is,  when  r<  '!i±l.     (See  §  182.) 
r  2 

If  ?i  is  odd  so  that  we  can  have  r  —  ^        ,  the  two  coefficients 


144  COLLEGE  ALGEBRA 

„(7m+i  and  „(7n-i  are  equal  and  greater  than  any  of  the  other 

2  2 

coefficients. 

If  n  is  even,  so  that  we  cannot  have  r  =  ^  "^    ,  the  greatest 

coefficient  is  ,,(7^,  where  r  is  the  greatest  integer   less  than 

Vl±1.,     This  integer  is  -. 

Hence  if  n  is  even,  JJn  is  the  greatest  coefficient  in  the  ex- 
pansion of  (a  +  hy. 

112.  The  expansion  of  (a  +  ^j*"  as  given  by  (I)  can  be 
described  in  words  as  follows : 

It  contains  (n  + 1)  terms. 

Tlie  first  term  is  a"  (or  a^'b^)  and  the  last  is  &"  {or  a°6"). 

The  second  term  is  na'^'^b. 

Tlie  exponent  of  a  in  any  term  after  the  first  is  one  less  than  it 
is  in  the  preceding  term. 

TJie  exponent  of  h  in  any  term  after  the  first  is  one  more  than 
it  is  in  the  preceding  term. 

TJie  coefficient  of  any  term  after  the  first  is  obtained  from  the 
preceding  term  by  dividing  the  product  of  the  coefficient  of  this 
term  and  the  exponent  of  a  by  one  more  than  the  exponent  of  b. 

It  should  be  observed  that  the  sum  of  the  exponents  of  a  and 
b  in  any  term  is  n. 

The  expansion  of  (a  —  by  can  be  obtained  from  that  of 
(a  +  by  by  putting  —  6  in  place  of  b,  since  a  —  b  =  a+(—b). 

Example  1.— Expand  /?^  +  ^V- 

Here  a  =  — ,  6  =  -,  and  n  =  6. 

5-4.S(2rY  (sy  5.4-3-2/2r\/sy     5  •  4  •  3  .  2  . 1 /j?\5 

■^      3!      Uj    Uj  4!         V  3  JVj)    "^           5!           UJ  ' 

^32r5      Wr^s  16r»8^     8  r V  ,  2  rs^  .     s^ 

243          81  136         225       375      3125* 


u 


PERMUTATIONS  AND  COMBINATIONS  145 

Example  2.  —Expand  (2  ic  —  3  r/)^. 
Here  a  =  2ic,  &=  —  3y,  and  n  =Q. 
Hence  (2  a:  -  3  yY 

=  (2a;)6  +  6(2a;)5(-3y)  +  ^  (2  a;)4(_  3  y)2 

=  64  a;6  _  576  r^?/  +  2160  cc^^a  _  4320  x^^/S  +  4860  icV  _  2916  xy^ 
+  729  2/6. 

EXERCISES 

Expand  the  following  expressions  by  the  binomial  theorem. 

1.  {x-^y)\  11.  {l+xy\ 

2.  (a-2y.  12.  (x^+y^f, 

3.  (3-0^)^  13.  (a +  6)* -(a -6/. 

4.  (6  +  iy.  14.  (5a6-«-2a;y/. 

5.  (2a  +  4cy.     \  15.  /'^  +  ^' 


19.    (m-^-n-y. 


10.  (.'c+ir. 


In  each  of  the  following  exercises  write  the  term  asked  for 
without  finding  any  other  term. 

21.  The  6th  term  of  (a  +  by. 

22.  The  6th  term  of  (a  -  by. 

23.  The  5th  term  of  (3  a  -  2  by. 
24L.  The  4th  term  of  (4  x^y  -  5  xy^^ 


iL.  ir  ^'^ 


146  COLLEGE  ALGEBRA 

25.  The  3d  term  of  (1  -  x)\ 

26.  The  8th  term  of  (2  x  +  —  V^* 

27.  Write  the  third  term  from  each  end  of  the  expansion 
of  (4a-7&)l  ' 

28.  Write  the  middle  term  of  (7  a^  -  4  Wf. 

29.  What  is  the  coefficient  of  x^  in  the  expansion  of  {1—xy^  ? 

30.  Write  the  first  five  terms  of  the  expansion  of  ( 1  -j-  - 

V      ^, 

31.  Are  the  coefficients  in  the  expansion  given  in  the  result 
of  the  first  illustrative  example  consistent  with  §  109  (I)  ? 

Compute  the  values  of  the  following  correct  to  two  decimal 
places : 

32.  (1.1)». 

Hint.— (1.1)8  =(1  +  . 1)8 

=  1  +  8(.l)  +  28(.1)2  +  56(.1)3  +  70(.l)*  +  56(.l)s 

+  28(.1)6  +  8(.1)-+(.1)8 
=  1  +  .8  +  .28  +  .056  +  .007  =  3.14. 

The  last  four  terms  have  been  neglected  since  they  can  have  no  influ- 
ence on  the  first  two  decimal  places. 


3^.    (3.1)9.  ^34 ,  (^99)6,  35,    (1.01)^  36.    (4.9)^ 


CHAPTER   XI 
MATHEMATICAL   INDUCTION 

113.  We  consider  in  this  chapter  a  method  of  proof,  com- 
mon in  mathematics,  of  a  certain  class  of  theorems  in  which 
an  integer  is  in  some  way  involved. 

We  shall  first  apply  the  method  in  question  to  a  simple 
theorem  and  shall  then  examine  its  essential  features. 

114.  Theorem. —  The  sum  of  the  first  n  even  integers  begin- 
ning with  2  is  equal  to  the  product  of  n  and  the  next  larger 
integer. 

I.  It  is  easy  to  verify  that  the  theorem  is  true  for  small 
values  of  n. 

For  example,  for  w  =  1,  we  have  2  =  1(1  -f-  1). 

II.  Suppose  that  the  theorem  is  true  for  7i  =  r ;  that  is, 
^^''^  2  +  4  +  6+  ...  +2r  =  r(r  +  l).  (1) 

Adding  2  (r  +  1)  to  each  member  of  this  supposed  equation, 
we  get 

2  +  4  +  6+  ...  +2r  +  2(r  +  l)=r(r  +  l)+2(r  +  l) 

=  (r  +  l)(r  +  2).  (2) 

Now  (2)  is  true  if  (1)  is  true.  But  (2)  is  merely  another 
form  of  the  statement  that  the  theorem  is  true  for  ?i  =  r  + 1. 
If  then  the  theorem  is  true  for  n  =  r,  it  is  true  for  n  =  r-\-l. 
But  we  know  from  I  that  it  is  true  for  7i  =  1,  and,  therefore,  it 
is  true  for  7i  =  1  + 1  =  2.  And  since  it  is  true  for  n  =  2,  it  is 
true  for  n  =  2  +  1  =  3.  We  can  proceed  in  this  way  by  suc- 
cessive steps  until  we  get  the  proof  that  the  theorem  is  true 
for  any  given  integral  value  of  n. 

147 


148  COLLEGE  ALGEBRA 

115.  This  method  of  proof  is  called  mathematical  induction. 
It  is  applicable  to  certain  theorems  in  which  a  positive  integer 
is  in  some  way  involved,  and  consists  of  two  parts ;  namely, 
the  direct  verification  of  the  theorem  for  the  smallest  admis- 
sible value  of  the  integer  and  the  proof  that  if  the  theorem 
is  true  for  one  value  of  the  integer,  it  is  true  for  the  next  greater 
value. 

116.  Necessity  of  both  parts  of  the  proof.  —  The  formula 

Z 
is  true  for  n  —  \.     It  is  also  true  for  71  =  2  and  n  =  3,  but  it  is 
not  true  for  all  values  of  n,  as  may  be  seen  by  putting  ti  =  4. 
This  gives 

12  I  22  -L  32  4.  42  =  80  —  28  +  4  ^  2g. 
2 
But  this  is  obviously  not  true. 
On  the  other  hand,  the  formula 

124.22+  ...  +n2  =  |n(7i  +  l)(2n  +  l)+5 

is  true  for  n  =  r-\-l,  provided  it  is  true  for  n  =  r. 
For,  if  the  formula  is  true  for  n  =  r,  we  have 

1^  +  22+. ..+r^  =  ^r(r  +  l)(2r  +  l)+5, 

and  if  we  add  (r  + 1)^  to  each  member  of  the  equation,  we  get 
12  +  22+. ..+r2+(r  +  l)2  =  ^r(r  +  l)(2r  +  l)+5+(r  +  l)2 

=  -J(r  +  l)(2r2  +  r  +  6r  +  6)  +  5 
=  i(^  +  l)(2r2+7r  +  6)  +  5 
=  K^  +  l)(^  +  2)(2r  +  3)  +  5 


But  this  is  what  the  original  formula  becomesjwlien  we  puj 
n  =  r  + 1.     Hence  if  the  formula  were  true  for  n  =  r,  it  would 
-^"be  true  for  n  =  r  +  1. 

J 


ih.^o  ^-rr 


MATHEMATICAL  INDUCTION  149 

That  this  argument,  however,  does  not  establish  the  validity 
of  the  formula  may  readily  be  seen  by  testing  the  formula  for 
n  =  l.     This  gives 

12  =  ^.  1(1  +  1)(2  +  1)  +  5  =  1 +  5  =  6, 

which  is  obviously  untrue. 

These  two  examples  show  that  both  parts  of  the  proof  as 
outlined  are  necessary  in  order  to  establish  the  truth  of  a 
theorem.  r^^ 

EXERCISES  .'W     ^     -V|A>i. 

Establish  the  :following  formulae  by  mathematical  induction : 

/l.   1  +  3  +  5+ ...+(2M-l)=7i2.  ^SL. 

^    \^     2.    12  +  22  +  32+ ...  +7i2=i-7i(n  +  l)(2n  +  l). 

3.  13_|_23  +  33H +n^  =  l7i''(7i-\-iy. 

4.  a  +  a?'  +  ar^  +  • . .  +  ar""^  =    ^    ~ — ^  • 

1  —  r 

5.  2  +  22  +  23+ ... +2"  =  2(2'' -1). 

^6.  JL  +  ^+...+      1 


1.22.3  n(n  +  l)      n-^1 

7.  2.4+4.6  +  6.8+  ...  +2n(2n  +  2)  =  ^(2n+2)(2n+4). 

o 

8.  1.2.3  +  2.3.4  +  3.4.5+  ...+n(n  +  l)(n  +  2) 

=  ^(n  +  l)(n  +  2)(n  +  3). 

9.  Prove  that  if  n  is  a  positive  integer,  a""  —  b""  is  divisible 
by  a  —  6. 

Hint,     cf+i  —  &'-+i  =  a(a^  -  b^)  +  b'-(a  —  b).      The  right  member  of 
this  identity  is  divisible  by  a  —  b,  if  a*"  —  ft*"  is  divisible  by  a  —  b. 

10.   Prove  that  if  n  is  a  positive  integer,  a^"  —  52™  ig  divisible 
by  a  +  6. 


150  COLLEGE  ALGEBRA 

117.   The  binomial  theorem  for  positive  integral  powers.  — 

The  student  is  familiar  with  the  fact  that 

(a  -\-by  =  a''-r2ab  +  h\ 
and  that  {a +  l>f  =  a^ +^  o?h -\-^  ah'' -[-b\ 

Now  these  relations  are  special  cases  of  a  general  theorem 
which  tells  us  how  to  write  out,  or  expand,  any  positive 
integral  power  of  a  binomial.  This  theorem  is  known  as  the 
binomial  theorem. 

It  says  that  if  n  is  any  positive  integer 

(fl  +  hy=a-+  na^-^b-\-  ^("~^) 0^-252^ 

« ! 

a(n-l)--(n-r+2)       +1     1 
(r-1)! 

_^_n(n-l)-(n-r+l)^„.,^  +  ...  +  ft., 
r! 

The  right  member  of  this  formula  can  be  described  as 
follows : 

Ji  contains  {n  -\-l)  terms. 

The  first  term  is  a""  (or  a" 6°)  and  the  last  is  6**  (or  a^6"). 

The  second  term  is  na^^'^h. 

Tlie  exponent  of  a  in  any  term  after  the  first  is  07ie  less  than  it 
is  in  the  preceding  term. 

The  exponent  of  b  in  any  term  after  the  first  is  one  more  than 
it  is  in  the  preceding  term. 

The  coefficient  of  a,ny  term  after  the  first  is  obtained  from 
the  preceding  term  by  dividing  the  product  of  the  coefficient 
of  this  term  and  the  exponent  of  a  by  one  more  than  the 
exponent  of  b. 

It  should  be  observed  that  the  sum  of  the  exponents  of  a 
and  b  in  any  term  is  n. 


MATHEMATICAL  INDUCTION  151 

118.  Proof  of  the  binomial  theorem. — We  shall  prove  the 
theorem  by  mathematical  induction. 

In  order  to  do  this  we  observe  in  the  first  place  that  the 
theorem  is  obviously  true  for  n  =  1.  (The  student  can  also 
verify  directly  that  it  is  true  also  for  n  =  2  and  n  =  3,  although 
this  is  not  necessary  for  the  argument.)  In  the  second  place 
we  assume  that  it  is  true  for  n  =  m* ;  that  is,  we  assume  the 
truth  of  the  statement  that 

(a  +  by  =  a-  +  ma^-^h  -h  .••  +  m(m- 1)  ■•♦  (?ft -r +  1)^„.-.^. 

+  •••  +  ma^'^-i  +  ?>"*. 

Then  we  multiply  each  member  of  this  identity  by  a  +  &. 
This  gives  us 

=  a'"+^+(m4-l)a"'6H —  -f-  ^ — ^^-^ — '^^ ^- — ^ ^J^-^a*"  '^+^6'^ 

t\ 

^ \-{m-\-l)ah'^-{-h'^^\ 

Now  this  expansion  for  (a  +  h)'^^'^  is  true  provided  that  the 
expansion  for  (a  +  6)'"  with  which  we  started  is  true.  More- 
over this  expansion  for  (a  +  6)"*+^  is  in  accordance  with  the 
binomial  theorem.  Hence,  if  the  theorem  is  true  for  n  ='m,\t 
is  true  for  n  =  m  -\-l.  But  we  have  already  observed  that  it  is 
true  for  n  =  1.  It  is  therefore  true  for  all  positive  integral 
values  of  n. 

In  deriving  the  expansion  of  (a  +  &)'"+^  we  made  use  of  the 
fact  that 

*  We  do  not  say  n=  r  because  we  are  here  using  r  for  another  purpose. 


152  COLLEGE  ALGEBRA 


m(m  — 1 

)...(m 

-r  + 

1)  ,  m(m- 

.  1). 

•••  (m- 

-r  +  2) 

_m(m  - 

r! 

-1).. 

•  (m  — 

?•  +  !)  +  ?'m(m 

-1)! 
-1).. 

•  (m  —  r 

+  2) 

7n(m  - 

-1)... 

•  (m  — 

r! 

7-  +  2)[(m 

—  r 

+  1)  +  rl 

rl 
_  (m  +  l)m(m  —  1)  « •  •  (m  —  r  +  2) 

119.   The  general  term.  —  The  term 

n(n-l)'"(n-r-{-l)  ^„_,^, 
rl 

is  the  (r  +  l)t7i  term  in  the  expansion  of  (a  +  ft)**.  It  is  called 
the  general  term  of  this  expansion  since  it  can  be  made  equal 
to  any  term,  except  the  first  one,  by  assigning  to  r  a  suitable 
value. 

The  expansion  of  (a  —  by  can  be  obtained  from  that  of 
(a  -h  by  by  putting  —  6  in  place  of  b,  since  a—b=a-{-(—b). 

For  exercises   on  the  binomial  theorem,  see  those  in  §  112. 


CHAPTER   XII 


COMPLEX   NUMBERS 


-  120.  In  considering  negative  numbers  for  the  first  time  it 
was  found  helpful  to  associate  with  every  point  of  a  given 
line  a  number  which  represented  its  distance  from  a  given 
point  (or  origin)  on  the  line.  In  order  to  distinguish  between 
two  points  equally  distant  from  the  origin  but  on  opposite 
sides  of  it  we  agreed  to  represent  the  distances  of  points  on 
one  side  of  the  origin  (say  the  right  side)  by  positive  numbers 
and  the  distances  of  points  on  the  opposite  side  of  the  origin 
by  negative  numbers.  The  numbers  that  represent  these 
points  were  called  real  numbers. 

Now  a  similar  but  more  extensive  association  of  num- 
bers with  points  will  serve  to  make  clear  in  a  simple  way  the 
essential  nature  of  a  more  complicated  kind  of  numbers  that 
are  of  great  importance  in 
mathematics;  namely,  the 
so-called  imaginary,  or  com- 
plex, numbers. 

Consider  two  perpendic- 
ular lines  X'X  and  Y'  Y 
that  intersect  at  0,  and  let 
the  real  numbers  represent 
the  points  of  X'X,  the  ori- 
gin being  taken  at  0. 

We  want  to  represent  the  points  of  the  plane  that  are  not  on 
X'X  by  symbols  which  we  can  use  in  a  way  that  resembles  as 
closely  as  possible  the  use  of  the  real  numbers  (the  symbols 
that  represent  the  points  of  X'X)  in  the  operations  of  addi- 
tion, subtraction,  multiplication,  and  division.  We  shall  call 
these  symbols  numbers. 

153 


Y 
5i+ 


Z-i- 


■51- 


154  COLLEGE  ALGEBRA 

121.  In  Chapter  V  we  represented  the  points  of  T'Y,  as 
well  as  those  of  X'X,  by  real  numbers,  but  for  our  present 
purpose  we  must  represent  the  points  of  the  plane  that  are 
not  on  X'X  by  numbers  different  from  the  real  numbers, 
since  every  real  number  represents  a  point  on  X'X,  and  we 
do  not  want  any  number  to  represent  two  points. 

The  number  by  which  we  shall  represent  the  point  on  Y'Y 
the  ^unit's  distance  above  0  will  be  called  i.  Any  point  on 
F'F  whose  distance  from  0  is  a  will  be  represented  by  the 
number  ia.  If  a  is  positive,  the  point  represented  by  ia  is 
above  0,  and  if  a  is  negative,  the  point  is  below  0.  Thus,  the 
point  represented  by  5^  (or,  it 5)  is  on  y F  five  units  above 
0,  and  the  point  represented  by  —5i  is  on  Y'Y  five  units 
below  0. 

If  we  should  mark  on  X'X  the  point  M  represented  by  the 
number  a  and  should  then  proceed  to  lay  off  the  point  ib  in 

the  way  just  described,  with  the 
exception  that  we  use  the  point 
M  instead  of  0  for  origin,  we 
should  get  the  point  P  which 
C  is  6  units  from  M  and  above  or 
below  it  according  as  b  is  posi- 
tive or  negative. 

We  shall  say  that  the  number 
representing  F  is  the  sum  of 
a  and  ib  and  shall  indicate  it  by  the  symbol  a  -\-  ib.  This 
gives  us  a  definition  of  the  sum  of  a  real  number  and  a  number 
of  the  form  ib  that  is  closely  analogous  to  the  definition  of  the 
sum  of  two  real  numbers  given  in  §  4. 

122.  Let  now  P  be  any  point  of  the  plane.  Draw  the  per- 
pendicular PM  from  P  to  X'X.  If  0M=  a  and  MP  =  b  the 
number  that  represents  P  is  a  4-  ib.  We  have  at  hand  there- 
fore a  system  of  numbers  that  will  represent  all  the  points  of 
the  plane,  and  conversely,  every  number  of  this  form ;  namely, 
a  -f  ib,  represents  a  point  of  the  plane. 


IT 


COMPLEX  NUMBERS 


155 


123.  Equal  numbers.  —  We  shall  agree  to  say  that  two 
numbers  are  equal  when  they  represent  the  same  point. 
Hence  if  a-\-ib  =  c-\-  id,  we  must  have  a  =  G,  and  h  =  d.  Con- 
versely, if  these  last  two  inequalities  hold,  the  numbers  a  +  ib 
and  c  +  id  are  equal. 

If  the  point  represented  by  a  +  ih  is  on  X'X,  6  :=  0.  If  the  point  is 
the  origin,  a  =  0,  &  =  0  ;  that  is,  if  a  +  ib  =  0,  a  =  0,  and  6  =  0. 


1.  Write  the 
that  represent  the  points 
designated  in  the  accom- 
panying figure. 

2.  Plot  the  points  repre- 
sented by  the  following 
numbers : 

3  +  1,  3  —  2*,   —3  +  1,  5i, 

—  ? ,  1  +  I,   —  1  —  3  I, 
4-22,     6,     -3  +  4?, 


EXERCISES 

numbers 


Y 

'D 

•B 

'C 

f 

0 

'E 

'G 

'F 

Y' 

X 


124.  It  was  stated  in  §  120  that  we  wanted  to  be  able  to 
use  these  new  numbers  in  a  way  resembling  as  closely  as  pos- 
sible the  use  of  real  numbers  in  the  operations  of  addition, 
subtraction,  multiplication,  and  division.  Now  the  essential 
point  in  the  use  of  real  numbers  is  that  it  is  governed  by  the 
assumptions  of  Chapter  I.  In  the  following  paragraphs  we 
shall  accordingly  describe  combinations  of  these  new  num- 
bers which  we  shall  call  addition,  subtraction,  multiplication, 
and  division  respectively,  and  which  will  also  be  governed  by 
these  assumptions.  It  will  be  seen  that  these  new  operations 
are  perfectly  natural  extensions  of  the  corresponding  opera- 
tions with  real  numbers.  We  have  already  described  the 
addition  of  a  real  number  and  a  number  of  the  form  ib. 


\fo 


156 


COLLEGE  ALGEBRA 


125.  Addition  and  subtraction.  —  By  the  sum  of  the  num- 
bers Xi  +  iyi  and  X2  +  iy2  representing  the  points  A  and  B  re- 
spectively, we  mean  the  number  x  +  iy  representing  the  point 
S  which  is  obtained  by  laying  oif  B  from  A,  instead  of  from 

0,  as  origin. 

Thus,  we  draw  ^iV  equal,  and 
parallel,  to  OM,  and  then  NS 
equal  and  parallel  to  3IB. 

The  student  can  readily  verify- 
that  S  is  the  fourth  vertex  of  the 
parallelogram  two  of  whose  sides 
are  OA  and  OB,  except  when  0, 
A,  and  B  lie  on  a  straight  line. 
He  can  also  see  that 


r 

^ ^!. 

r' 

/^' 

-^           0 

M 

r' 

Hence, 


x  =  Xi  +  X2  and  y=yi-\- 
{xi  +  iy{)  +  {X2  +  iy2)  =  (xi  +  x^)  +  i{yx  +  yo). 


By  the  difference  of  x\  +  iy\  and  x<2,  +  i?/2,  or  (a;i  +  iy{)  —  {xi  +  2*2/2) » 
we  mean  the  number  x  -f-  i?/  such  that 

y^\  +  «yi  =  (a;  +  iV)  +  {^1  +  *y2) 
=  (x  +  Xi)  +  i(y  +  ^2) . 

It  follows  from  §  123  that 

xx^x^-xi  and  yi  =  y  +  y2' 
Hence,  x  =  Xi  —  x^  and  y  =  yi  —  yii 

and  therefore  {xi  +  i>i)  -  (^2  +  12/2)  =  (a*!  -  ajg)  +  i(2/i  -  2/2). 

This  difference  can  be  obtained  graphically  by  forming  the  sum  of 
xi  +  iyi  and  —  X2  +  i(-  2/2). 


EXERCISES 

In  each  of  the  following  exercises  represent  graphically  the 
numbers  in  the  parentheses,  and  their  sum  or  difference,  as  the 
case  may  be : 

1.  (i  +  2?:)H-(5-5i).  4.   3-(6-2r). 

2.  2i-4i.  5.   3  2^-2. 

3.  (i_2i)-(2  +  50.  6.    (6  +  20  +  (5-4i). 


COMPLEX  NUMBERS 


157 


7.  (7-i)-(7-f-30. 

8.  (5-40  +  (5-40. 

9.  (_3_3t)+4i. 

10.  3 -(6 -6/). 

11.  (8+4i7-(8  +  4*). 

12.  (_3i+2i)-(-3i+i). 


13.  (V2-hV3  0  +  (V2-V3  0. 

14.  (4^  +  3)  +  (4  +  3^). 

15.  (2  +  6  i)- (2  +  60- 

16.  (3-2*)  +  (2i-3). 

17.  (7-2i2  +  (7  +  20.    _ 

18.  (l+V5i)  +  (2  +  2V5i). 


19.  What  is  the  relation  between  the  points  represented  by 
3  +  4  I  and  —  3  —  4 1'  respectively  ?  Plot  these  points  and  also 
the  points  3  —  4 1  and  —  3  +  4 1. 

20.  Under  what  circnmstances  is  the  sum  of  two  complex 
numbers  a  real  number  ? 

21.  Apply  the  definitions  of  addition  and  subtraction  in 
this  article  to  two  real  numbers  and  compare  the  results  with 
those  obtained  by  applying  the  definitions  of  §§  4  and  6. 


126.  Multiplication.  — By  the 
product  of  I  and  the  number 
representing  any  point  P  we 
mean  the  number  representing 
the  point  to  which  P  is  brought 
by_reyobdiig-jftP  countgrclock- 
wise  around  0  through  an  angle 
of  90°. 


A'^-+- 


^B2 


Y' 


Thus,  if  the  angle  POQ  is  a  right 
angle  and  OP  =  OQ,  the  number 
representing  Q  is  equal  to  the  num- 
ber representing  P  multiplied  by  i. 

In  the  figure  let  OBx  =  OB2  = 
OBs.  If  b  represents  the  point  J5i  on 
X'X,  the  product  of  i  and  b  repre- 
sents B2  on  Y'  Y.  But  we  have  al- 
ready represented  this  point  by  ib.^ 
Hence  we  say  that  ib  is  the  product  ' 
of  i  and  b. 


158  COLLEGE  ALGEBRA 

The  product  of  i  and  ih  represents  the  point  B^  on  X'X.     But  -  h 
also  represents  this  point.     Hence, 

i  •  ib  =—  b, 
or  i^b=-b. 

This  is  equivalent  to  saying  that^, 

1^""'^^  1. 


127.  AVhen  we  consider  that  the  product  of  two  positive,  or 
two  negative,  numbers  is  positive,  it  seems  strange  that  we 
can  have  a  number  like  i  whose  square  is  a  negative  number ; 
or,  in  other  words,  that  we  can  have  a  square  root  of  a  negative 
number.  The  explanation  is  that  this  number  is  neither  posi- 
tive nor  negative,  since  it  represents  a  point  not  on  the  axis 
X'X  It  is  a  new  kind  of  number  and  it  ought  not  to  be 
surprising  that  it  should  have  properties  not  possessed  by  the 
old  numbers. 

If  we  had  never  heard  of  negative  numbers,  it  would  seem 
just  as  strange  to  talk  about  a  number  that  is  equal  to  3  minus 
5  as  it  does  to  talk  about  a  number  whose  square  is  —  1. 

128.  Definitions. — Numbers  of  the  form  ib,  where  &  is  a 
real  number,  different  from  zero,  are  called  pure  imaginaries. 
Numbers  of  the  form  a  -\-  ib,  where  a  and  b  are  real  numbers 
and  b  is  not  equal  to  zero,  are  called  imaginary  numbers. 

This  term,  imaginary,  was  applied  at  a  time  when  it  was 
supposed  that  these  numbers  had  no  existence  and  were  not 
real.  But  with  the  conception  of  numbers  that  has  been  set 
forth  here  we  see  that  they  are  just  as  real  as  the  points  of  a 
plane,  which  they  represent,  and  therefore  just  as  real  as  the 
so-called  real  numbers. 

Since  imaginary  numbers  are  formed  by  a  combination  of 
two  essentially  different  kinds  of  units ;  namely,  1  and  i,  they 
are  more  properly  called  complex  numbers. 

In  the  complex  number  a  +  ib,  a  is  called  the  real  part  and 
ib  the  imaginary  part. 


COMPLEX  NUMBERS  159 

In  a  real  number  the  imaginary  part  is  zero  and  in  a  pure 
imaginary  the  real  part  is  zero. 

Complex  numbers  that  differ  only  in  the  sign  of  their  imagi- 
nary parts  are  said  to  be  conjugate  to  each  other. 

Thus  4  +  3  I  and  4  —  3  i  are  conjugate,  as  are  also  a  4-  ih  and  a  —  ib. 

A  real  number  is  its  own  conjugate. 

129.  Product  of  two  complex  numbers.  —  We  are  now  in 

a  position  to  define  what  we  mean  by  the  product  of  two  com- 
plex numbers  like  a  +  ib  and  c  +  id. 
We  agree  to  say  that 

(a  +  ib)  (c  +  id)  =  ac  +  ibc  +  iad  +  i'^bd  =  (ac  —  bd)  +  i(bc  +  ad). 

It  will  be  observed  that  this  product  is  just  what  it  would 
be  if  the  numbers  a,  b,  c,  d,  and  i  were  all  real,  with  the  excep- 
tion that  it  makes  use  of  the  fact  that  v^  =  —  1. 

130.  Theorem.  —  The  sum  and  the.  product  of  two  conjugate 
complex  numbers  are  real  numbers. 

For  {a  +  ib)  +  (a  —  ib)  =  2  a, 

and  {a  +  ib)  (a  -  ib)  =  a'^ -^  VK 

131.  Division  of  complex  numoers. — -We  define  the  division 
of  a  +  ib  by  c  -h  id  to  be  the  operation  of  finding  the  number 
x  +  iy  such  that  a-\-ib  =  {x-\-  iy)  (c  +  id). 

The  indicated  quotient  .    can  be  changed  to  the  form 

X  -\-  iy  by  multiplying  both  the  numerator  and  the  denominator 
of  the  fraction  by  the  conjugate  of  the  denominator.     Thus 

a  4-  ib  _  (a  -f  ib)(c  —  id)  _  (ac  +  bd)  +  i  (be  —  ad) 
c  +  id~  (c  +  id) (c  —  id)  ~  c^  +  d^ 

_  ac  -f-  bd  .    .  be  —  ad 

This  process  justifies  the  statement  of  Assumption  XII  of 
Chapter  I  as  applied  to  complex  numbers. 


160  COLLEGE  ALGEBRA 

132.  We  have  now  defined  the  four  fundamental  operations 
of  addition,  subtraction,  multiplication,  and  division,  as  applied 
to  complex  numbers,  in  a  way  which  can  be  shown  to  be  con- 
sistent with  the  assumptions  of  Chapter  I.  Moreover  when 
applied  to  numbers  whose  imaginary  parts  are  zero,  these 
operations  are  the  same  as  the  operations  of  addition,  subtrac- 
tion, multiplication,  and  division  respectively,  as  applied  to 
real  numbers. 

133.  Principal  square  root  of  negative  numbers.  —  Since  i^ 
and  (—  if  are  both  equal  to  —  1,  i  and  —  i  are  square  roots  of 

—  1.  We  shall  call  i  the  principal  square  root  of  —  1  and  rep- 
resent it  by  V—  1.     Thus  i  =  V—  1. 

By  the  principal  square  root  of  any  negative  number  such  as 

—  a,  where  a  is  a  positive  number,  we  mean  the  product  of  i 
and  the  principal  square  root  of  a  (see  §  55).  We  shall  use 
the  symbol  V  —  ct  to  indicate  this  principal  square  root. 

Thus  V—  a  =  y/a  •  i  =  Va  V—  1. 

If  b  is  also  a  positive  number,  then 

V—  a  •  V—  b  =  ■\/a  •  i  •  Vb  •  i  =  Va6  .^—  —  y/ab. 

The  student  is  warned  against  saying 

V—  a  .  V—b  =  V—  a  •  --  b  =  Vab. 

EXERCISES 

Reduce  each  of  the  following  expressions  to  the  standard 
form  a  -\-  ih. 

1.  (3-2^)(4-5^).  ^    1 

2.  (2  +  1)2.  '    i' 

3.  i'.  Jidi  VI  i  p4M  ^y^S.    (2  +  V^(7  -  V~i). 
4-    **•  djJvi^   ivY***^  P    Hint. 

5.  i\     k .;;  iJi    S^  .  A'-*'^  .         2  +  V^l  =  2  +  V3  I, 

6.  ^^^  |i'  >^x  |#^  '^    h^  ^^Ljr^^/~^  =  'J-2L 


COMPLEX  NUMBERS  161 


18.   5  *  .  —  8  i. 


3+i       2+i 
2  ^  ^       3_^,.      2  +  1 


12. 


^     __  .  ^,     1-V3i     -1  +  V3"i 

1  +  V3i         /  ^^-  2  2 


13.    (2-0(2  +  i)(3  +  2i).  22^    V-7+V-8 


14     1  +  ^' 


V-9+V-2 


1-^-  23.  ^ 


15.  (V2  4-V5^)(lV2^-V5). 

16.  (V^  +  V^i)(^V^  +  V^). 


V3  +  V2i 


25.  What  is  the  relation  between  the  points  represented  by 
two  conjugate  numbers  ? 

Solve  the  following  equations,  plot  the  points  represented 
by  the  roots,  and  show  graphically  that  the  sum  of  the  roots 
of  each  equation  is  zero. 

26.  ar5-l  =  0.  28.    ic*-l  =  0. 

27.  0)3+1  =  0.  29.    x«-l  =  0. 

30.  Show  that  1  -f  i  is  a  root  of  the  equation 

31.  Show  that  ^  —  ^i  is  a  root  of  the  equation 

36  a^.  +  36  a;2  -  47  a;  +  50  =  0. 


162 


COLLEGE  ALGEBRA 


134.  The  polar  representation  of  complex  numbers.  —  There 
is  another  method  of  representation  of  complex  numbers  that 

is  sometimes  useful.  Consider 
the  complex  number  x-\-iy  that 
represents  the  point  P  of  the 
figure. 

If  OP=r  and  Z.X0P=6, 

then  X  =  r  cos  6,        (1) 

y  =  r  sin  6,        (2) 

^  +  f  =  A  (3) 

and  therefore 
x-{-iy  =  r  cos  6  +  ir  sin  d  =  r  (cos  $  -\-i  sin  6). 

This  representation  of  a  complex  number  in  terms  of  r  and  6 
is  called  the  polar  representation  of  the  number.  The  angle  9 
is  called  the  amplitude,  or  argument,  and  the  distance  OP,  or  r, 
is  called  the  modulus,  or  absolute  value,  of  the  complex  number. 

It  follows  from  formula  (3)  that 


r  =  -yjx^  +  2/^ 


is  the  modulus  of  the  number  x  +  iy.     It  is  always  considered 
positive. 

It  follows  from  (1)  and  (2)  that  the  argument  6  oi  x-\-  iy  is 
given  by  the  formulae 


iC 


x 


0  =  cos~^-=  cos~^ 


=  sin~^^  =  sm~^ ^ 


■y/x^  -j-  2/2 


For  example,  the  modulus  of  2  —  3  lis  r  =  VlS  and  the  argument  is 

5^' 


2  3 

d  =  cos-i =  sin-i  — - 


Vis 


COMPLEX  NUMBERS  163 

135.  Theorem.  —  Tlie^  modulus  of  the  product  of  two  numbers 
is  equal  to  the  product  of  their  moduli  and  the  argument  of  the 
product  is  equal  to  the  sum  of  their  arguments. 

For 

r(cos  6-{-i  sin  0)  •  7*'(cos  6'  +  i  sin  0') 
=  {r  cos  6  +  ir  sin  ^)(/  cos  0'  +  ii-'  sin  6') 
—  rr'  cos  6  cos  6'  —  rr'  sin  6  sin  6' 

+  i(r/  sin  6  cos  6'  +  r/  cos  6  sin  6') 
=  r/[(cos  6  cos  ^'  —  sin  6  sin  ^')  +  i(sin  0  cos  ^'H-  cos  ^  sin  $')'] 
=  r/[cos((9  +  ^')  +  i  sin  ((9  +  6')\ 

136.  Since 

r(cos  ^  +  i  sin  ^)  _  r(cos  6  -\-i  sin  ^)(cos  ^^  —  ^^  sin  6^) 
/(cos  ^'  +  i sin  6')  ~  /(cos  ^  +  i  sin  ^')(cos  6'—  i  sin  6*') 

_  r[cos(^-^-)+.sin(^-^0]  ^  .^  ^       ^   _   ,  .    ^   _   , 

/(cos^^'+sin^^')  /*-       ^  ^^  ^  ^-'' 

we  have  the 

Theorem.  —  The  modulus  of  the  quotient  of  two  numbers  is 
equal  to  the  quotient  of  their  moduli  and  the  argument  of  the 
quotient  is  equal  to  the  argument  of  the  dividend  minus  the  argu- 
ment of  the  divisor. 

137.  De  Moivre»s  Theorem.  — If  the  two  factors  in  §  135 
are  equal,  we  have 

[r (cos  e  +  i  sin  0)^  =  r\cos  2  0  +  i  sin  2  0). 

This  is  a  special  case  of  the  following  more  general  formula 
[r(cos  6  -\-  i  sin  0)y  =  r"(cos  nd  -f  i  sin  nO).  (I) 

The  statement  of  this  relation  is  known  as  De  Moivre's  Theo- 
rem. In  the  case  that  ti  is  a  positive  integer  this  theorem  can 
be  proved  by  means  of  mathematical  induction  in  connection 


164  COLLEGE  ALGEBRA 

with  the  theorem  of  §  135.     The  details  of  the  proof  are  left  to 
the  student.     He  should  go  through  them  with  great  care. 

The  proof  of  (I)  when  n  is  the  reciprocal  of  a  positive  in- 
teger, say  — ,  is  as  follows  : 
m 

Let  e  =  TYKJ). 

Then 

ill 

(cos  e+i  sin  d)m  =  (cos  m<p  +  i  sin  m<p)m  =  [(cos  (p  +  i  sin  0)»»]m 

=  cos  0  +  i  sin  0  =  cos— ^  +  i  sin  —  6. 
m  m 

ill  1 

Hence,         [r(cos  6  +  i  sin  d)']m  =  rm(co&-e  +  i  sin  — ^). 

m  m 

Formula  (I)  also  holds  for  all  rational  values  of  n. 
138.   Roots  of  numbers.  —  Suppose  that 

Then  by  formula  (I)  of  the  preceding  article,  provided  that 
m  is  any  integer, 

P^cos  0_±m_m^  ^  .  ^.^  e  +  rr^VY 

=  r  [cos(^  +  m  .  360°)  +  i  sin(^  +  m  •  360°)] 
=  r  (cos  6-\-i  sin  0)z=zx-\-%y. 

It  should  be  remembered  that  rn  represents  the  principal  nth 
root  of  r. 

Hence  any  number  of  the  form 

^fcos ttJ!Lz3^  +  i  sin e+jn,Zm,  (l) 

\  n        '  n  J 

where  m  is  an  integer,  is  an  nth  root  of 

r(cos  6-^i  sin  6),  or  x  +  iy. 

If  m  is  any  integer  (positive  or  negative),  and  we  divide  m 
by  n,  we  shall  get  a  quotient  q  and  a  remainder  s.  It  may  be 
that  s  will  be  zero,  but  in  any  case  we  can  take  q  so  that  s  will 


COMPLEX  NUMBERS  165 

be  less  than  n,  and  not  less  than  0.     Then 

w,  =  qn-[-  s, 

and    ^  +  ^-^^Q°^^  +  (g^  +  ^)^^Q°^^  +  ^-^^Q°  I  Q,360° 
n  n  n 

But  we  know  from  trigonometry  that  two  angles  which  differ 
by  a  multiple  of  360°  have  the  same  sines  and  cosines.    Hence 

cos  (  — 1-  q  •  ooO     =  cos  — ' ) 

\  n  J  \         '>^  J 

.           •    fe-{-S'  360°  ,        oaf^o\        •   fO-{-S'  360°\ 
and         sm    —^ h  q  •  360     =  sm  — — —  )• 

\         n  J  \         f^         J 

If  mi  and  mg  are  any  two  values  of  m  and  if 

cos  ^±^360:  ^ ^^^  g  +  m,360°^^^  ^.^  6±m,S^ 
n  n  n 

.    i9  +  m2  360°  ,,        ^  + mi 360°  -..^      .         ^  +  m2  360° 

=  sm  — ■ — ,  then  — ' — can  diner  from  — ' — ■ 

n  n  n 

only  by  a  multiple  of  360°,  since  two  angles  which  have  the 
same  sines  and  cosines  can  differ  only  by  a  multiple  of  360°. 

But  this  difference  is  ^^i  ~  '^v ^j^^j  ^g  j^q^.  ^^  multiple  of 

n 
360°  when  mi  and  mg  are  unequal  integers  less  than  n  and  not 
less  than  0. 

Therefore  the  following  values  of  m  give  all  the  distinct  num- 
bers of  the  form  (1) :  0, 1, 2,  •  •  -,  n  —  1.    This  proves  the  following 

Theorem.  —  Jf  n  is  a  positive  integer,  any  number  different 
from  zero  has  exactly  n  distinct  nth  roots. 

Since  all  the  nth  roots  have  the  same  modulus,  and  since  the  \ 

360°   •  \ 

argument  of  any  one  of  them  increased  by  is  the  argu- 

n 

ment  of  another  one,  we  see  that  the  points  represented  by  the 
nth  roots  of  a;  +  iy  all  lie  on  the  circumference  of  a  circle 
whose  center  is  at  the  origin  and  divide  this  circumference  into 
n  equal  parts. 


166  COLLEGE  ALGEBRA 

EXERCISES 

Find  the  modulus  and  the  argument  of  each  of  the  following 
numbers  and  plot  the  number. 

1.  2  +  5/.  '        3.  6-V^~2.  5.  2  +  L 

2.  -3 1.  4.  8.  6.  l-^. 

'■'■    _  "l^:- 

8.  (7  +  V-10)(3+V"^=^).       11-  5  (cos  30° +  1  sin  30°). 

Simplify  the  following  products  : 

13.  2  (cos  10°  + I  sin  10°)  •  5  (cos  25°  + 1  sin  25°). 

14.  -  5  (cos  30°  +  i  sin  30°)  •  (cos  15°  +  ^  sin  15°). 

15.  (1  +  if.  19.  [i(cos  180°  +  i  sin  180°)]^. 

16.  7if^-\-^\  20.  [3(cos60°  +  isin60°)]0 
,,    /I  _'iV3VV2^  .V2X        ''•  (-«30°  +  .-sin30°)3. 

\2        2   j\  2  2  J        22.  [2 (cos 40° +  ^  sin 40°)]'. 

18.  [4(cos20°+isin20°)]^         23.   [4(cosl20°  +  isinl20°)]3. 

24.   Find  all  the  cube  roots  of  1  +  i. 
Here  r  =  Vl  +  1  =  V2, 

sin  ^  =  cos  ^  = 

V2 
Hence,  6  =  45°, 

and  1  +  I  =  V2  (cos  45°  +  *  sin  45°) . 

We  shall  get  all  the  cube  roots  of  1  +  i  by  giving  to  m  in  this  expres- 
sion the  successive  values  0,  1,  2. 

For  m  =  0,  we  get  v^2(cos  15°  +  isin  15°). 
For  m  =  1,  we  get  v^2(cos  135°  +  isin  135°). 
For  w  =  2,  we  get  v/2(cos255°  +  isin  255°). 


COMPLEX  NUMBERS  167 

25.  Find  and  plot  all  the  cube  roots  of t^  +  ^-tt— 

26.  Find  and  plot  all  the  fourth  roots  of  \^' 

1  —  ^ 

27.  Find  and  plot  all  the  cube  roots  of  —  i. 

28.  Solve  the  equation  ar'  —  32  =  0,  and  plot  the  roots. 
Here  x^  =  32. 

Now  32  =  32 (cos  0°  +  i  sin  0^). 

TT  6/sH      a/        0°  +  m  •  360°  ,    .   .    0°  +  m  •  360°\ 

Hence  v32  =  2   cos      ^ h  \  sin  — ^^—^ ), 

V  5  5  / 

where  m  =  0,  1,  2,  3,  4. 

Hence  the  roots  are  2,     2  (cos  72°  +  i  sin  72°) , 

2(cos  144°  +  i  sin  144°), 
2(cos216°-f  isin216°), 
2(cos288°  +  isin288°). 

Solve  the  following  equations  and  plot  the  roots: 

29.  a^  -  1  =  0.  30.    a;*'  +  1  =  0. 

Compare  the  method  of  solution  used  in  Exs.  29  and  30  with 
that  used  in  Exs.  26,  27,  §  133. 

31.    a;*-16  =  0.         32.    x'-X^^.         33.    aj3  +  27  =  0. 

34.   What  is  the  relation  between  the  roots  of  Equation  30 
and  those  of  Equation  33  ? 

Simplify  each  of  the  following  quotients  : 


\-i 


3_2V-4 
^^     V2J-^V2^  ^^    6(cos60°-f  isin60°) 

V3H-t  '   3(cos20°  +  *sin20°)' 

1 
41. 


cos  40°  -  i  sin  40° 


CHAPTER  XIII 
THEORY   OF  EQUATIONS 

139.  The  student  is  familiar  with  the  solution  of  equations 
of  the  first  and  second  degrees  in  one  unknown;  that  is,  he 
knows  how  to  identify  numbers  described  by  such  equations. 

It  happens,  however,  that  many  of  the  descriptions  met  with 
in  the  applications  of  mathematics  are  in  the  form  of  equations 
of  higher  degrees,  and  it  is  far  more  difficult  to  identify  num- 
bers from  such  descriptions  than  from  linear  or  quadratic  equa- 
tions. We  shall  make  no  attempt  to  find  all  the  roots  of  such 
equations,  but  shall  confine  our  attention  to  the  problem 'of 
finding  their  real  roots. 

140.  This  latter  problem  naturally  divides  itself  into  two 
parts : 

1.  The  problem  of  finding  the  rational  roots. 

2.  The  problem  of  finding  the  approximate  values  of  irra- 
tional real  roots. 

This  is  the  most  important  problem  considered  in  this  book 
and  the  purpose  of  this  chapter  is  to  show  how  to  solve  it. 
Everything  given  in  this  chapter  has  a  direct  bearing  on  this 
solution. 

141.  We  shall  use  such  symbols  as  f(x),  F(x),  and  Q(x)  to 
represent  polynomials  in  x.  These  symbols  are  read  "/of  x," 
"  large  F  of  a?,"  and  "  Q  oi  x  "  respectively. 

If /(a?)  stands  for  afpc^'-^a^x*'-^  -{-•.•  +«„_!« +  a„;  that  is,  if 

f(x)  =  aoaJ**  +  aia;""^  4-  •••  +  a«-ia;  +  «„, 

we  shall  use  the  symbol  /(c)  to  stand  for  the  number  obtained 
by  putting  c  for  x  in  this  polynomial. 

168 


THEORY  OF  EQUATIONS  169 

Thus  if  f(x)=^x^  +  x^-5x  +  2, 

then  /(2)  =4  .  23  4-  22  -  5  .  2  +  2  =  28, 

and  /(O)  =4.0  + 0-5.0 +2  =  2. 

EXERCISES 

Determine  7i  and  «„,  ctj,  ••• ,  a„in  each  of  the  following  cases : 

1.  f(x)=-2x*-\-7a^-Sx^-5x-6. 

2.  f(x)  =  x^-{-x'-l. 

3.  f(x)  =  5x^  +  3a^-\-10x. 

4.  f(x)  =  x^-x*-\-2a^  +  5. 

5.  f(x)=x'-l. 

6.  f(x)  =  (x-i-l)(x  +  2)(x  +  S), 

7.  In  each  of  the  preceding  exercises  find /(I),  /(—  2),  and 
/(O). 

142.  Remainder  Theorem.  —  When  f  (x)  is  divided  by  x  —  r 
the  remainder  isf{7-). 

This  remainder  must  be  a  constant  since  it  is  of  lower  de- 
gree in  X  than  the  divisor,  which  is  of  the  first  degree.  The 
quotient  must  be  a  polynomial  in  x.  We  shall  represent  it  by 
Q(x)  and  the  remainder  by  E. 

Then  f(x)  =  Q(x){x  -r)-\-  R. 

If  in  this  identity  we  put  x  =  r,  we  get 

/(r)  =  Q{r)(r  -  r)  +  R. 

But  Q(r)  (r  —  r)  =  0,  since  r—r  =  0. 

Hence  R  =  f(r). 

The  student  should  observe  that  the  expression  for  B  does  not  contain 
X  and  that  therefore  B  is  the  same  number,  no  matter  what  value  is  given 
to  X. 

Corollary.  —  If  r  is  a  root  of  the  equation  fix)  =  0,  then  x  —  r 
is  a  factor  of  fix),  and  conversely. 

For,  by  the  theorem,  the  rehiainder  obtained  in  dividing /(ic) 
by  a;  —  r  is  f{r),  and  by  hypothesis  /(r)  =  0. 


170  COLLEGE  ALGEBRA 

Conversely,  if  f(x)  =  Q  (x)(x  —  r),  then  f(r)  =  Q  (r)  (r  —  r)  =  0, 
and  r  is  a  root  of  the  equation  f(x)  =  0.     (See  §  65.) 

This  corollary  is  sometimes  referred  to  as  the  Factor 
Theorem.  i 

EXERCISES 

Perform  the  following  divisions  and  check  your  work  by 
means  of  the  Remainder  Tlieoi-em. 

1.  Q^-^^.x'^  +  ^x-lOhj  x-2, 

2.  2  ic^  —  7  a^  —  a;  +  4  by  a;  -t  1. 

3.  a?*  +  a^  +  a;2  +  a;  +  1  by  cc. 

4.  7a;^  +  2x^  +  8a;3  — 6ar^  — 5a;  +  4.by  a^— 1. 

5.  a;^  +  1  by  a;  +  3-  6.    a;^  —  ar^  +  1  by  a;  —  1. 

7.  a^-6a?2-3a;  +  2by  a;+.5. 

8.  3 a^  +  5 a^- 6a; -2  by  a; +  2. 

143.  Synthetic  division.  —  In  looking  for  the  roots  of  the 
equations  considered  in  this  chapter  we  shall  have  frequent 
occasion  to  find  the  values  of  polynomials  in  x  for  given  values 
of  X.  The  Remainder  Theorem  tells  us  that  these  values  are 
equal  to  the  remainders  obtained  by  dividing  the  polynomials 
by  expressions  of  the  form  x  —  r.  It  is  therefore  important 
to  be  able  to  find  these  remainders  as  quickly  as  possible. 

Consider  the  process  of  dividing  2a:^— 5a?^4-3a;  —  4bya;  —  3. 

2a^-5a;2+3a;-4|a;-3 


2:x?-Qx'  2ar2+a;4-6 


ar^-f  3  a; 

X?- 

-3a; 

6a;- 

-   4 

6a;- 

-18 

14 

If  we  wish  to  shorten  the  work  of  this  division  as  much  as 
possible,  we  observe  that  it  would  be  sufficient  to  write  merely 
the  coefficients  of  the  various  powers  of  x  in  the  dividend  and 


THEORY  OF  EQUATIONS  171 

the  quotient  and  that  we  could  omit  the  first  term  of  the 
divisor  without  causing  confusion,  since  all  the  divisors  we  are 
considering  have  the  same  first  term.  Moreover  only  the  first 
term  of  each  partial  remainder  need  be  brought  down.  Then 
our  work  could  be  arranged  as  follows : 

2-5  +  3-41-3 


2-6  2+1+6 

1 

1-3 
6 

6-18 
14 

We  note  further  that  it  is  unnecessary  to  write  the  coef- 
ficients of  the  quotient,  since  these  are  equal  respectively  to 
the  first  (or  left  hand)  coefficients  of  the  dividend  and  the 
successive  remainders.  The  coefficients  of  the  first  terms  of 
the  successive  partial  products  may  also  safely  be  omitted. 
The  arrangement  of  the  work  will  then  be  as  follows : 

2-5  +  3-41-3 
-6 
1 
-3 
6 
-18 
14 

We  could  put  3  in  place  of  —  3  if  we  remember  to  add  the 
partial  products  instead  of  subtracting  them.  It  is  desirable 
to  do  this  because  what  we  are  really  trying  to  find  is  the 
value  of  the  polynomial  when  3  is  put  in  place  of  x.  More- 
over all  the  partial  products  can  be  written  on  one  line.  The 
work  can  therefore  be  arranged  as  follows : 

2-5  +  3-    4|3^ 

+  6  +  3  +  18 
2  +  1+6  +  14 


172 


COLLEGE  ALGEBRA 


The  successive  terms  in  the  last  line  reading  from  left  to 
right,  except  the  last  one,  are  the  coefficients  of  the  descending 
powers  of  x  in  the  quotient.     The  last  term  is  the  remainder. 

Thus  if  we  divide  2x3— 5aj2  +  3aj  —  4  by  x  —  3  we  get  the  quotient 
2x2  + X +  6  and  the  remainder  14.  Hence  this  polynomial  equals  14 
when  3  is  put  for  x. 

This  shortened  form  of  division  is  known  as  Synthetic 
Division. 

144.  Rule  for  synthetic  division.  —  In  order  to  divide  the 
polynomial  f(x)  by  x  —  r,  arrange  f{x)  according  to  descending 
powers  of  x.  If  f(x)  =  a^x'^  +  a^x^''^  +  •••  +  a„_ia;  -f  a„,  supply 
zeros  in  the  places  of  the  coefficients  of  the  missing  terms  and 
write  the  successive  coefficients  in  a  row  beginning  with  aQ. 

Bring  down  aQ,  multiply  it  by  r  and  add  the  j^roduct  to  a^ ;  mul- 
tiply the  resulting  sum  by  r  and  add  the  product  to  a^.  Continue 
this  process  of  multiplying  each  sum  by  r  and  adding  the  product 
to  the  next  coefficient  until  a  product  has  been  added  to  the  last 
coefficient.     The  last  resulting  sum  is  the  remainder. 

The  numbers  in  the  roiv  of  sums,  except  the  last  one,  are  the 
coefficients  of  the  successive  terms  of 
the  quotient  beginning  with  the  term 
containing  x""'^. 

145.  Application  of  synthetic  di- 
vision. —  This  method  of  finding  the 
value  of  f{x)  for  a  given  value  of  x 
can  be  used  to  good  advantage  in 
finding  points  on  the  loci  of  equa- 
tions of  the  form  y  =  f{x).  If 
f(a)  —  b,  the  point  {a,  b)  is  on  the 
locus  of  the  equation  y=f(x).  Thus  the  locus  of  y=x'^—6x-{-b 
passes  through  the  points  (0,  5),  (1,  0),  (2,  —  3),  etc. 

146.  Graphical  solution  of  equations.  —  The  abscissae  of  the 
points  that  are  common  to  the  locus  of  the  equation  y=f(x) 
and  the  ii?-axis  are  the  real  roots  of  the  equation  f(x)  =  0.    This 


THEORY  OF  EQUATIONS 


173 


suggests  at  once  that  we  can  find  the  real  roots  oif(x)  =  0  graphi- 
cally by  drawing  the  locus  of  the  equation  y=f(x)  and  measur- 
ing the  abscissae  of  the  points  common  to  the  locus  and  the  a;-axis. 

If  the  degree  of  f(x)  is  greater  than  2,  the  computation 
necessary  in  order  to  draw  the  curve  enables  us  to  make  a  fair 
approximation  to  the  roots  without  the  figure.  Nevertheless, 
a  consideration  of  the  figure  often  gives  us  important  informa- 
tion concerning  the  roots. 

This  is  illustrated  in  the  proof  of  the  following 

Theorem.  —  If  a  and  h  are  real  numbers  and  f(a)  and  f(b) 
have  opposite  signs,  there  is  at  least  mie  real  root  of  the  equation 
f(x)  =  0  between  a  and  b. 

If /(a)  and/(6)  have  opposite 
signs,  the  locus  of  the  equation 
y  =f(x)  is  on  opposite  sides  of 
the  ic-axis  at  the  points  whose 
abscissae  are  a  and  b  respec- 
tively. 

Now  this  locus  is  an  unbroken 
curve  and  does  not  turn  back  on 
itself  (see  figure),  since  for  each  value  of  x  there  is  only  one 
value  for  y,  and  therefore  it  must  cross  the  a>axis  at  least  once 
between  the  point  whose  abscissa  is  a  and  the  point  whose 
abscissa  is  b.     This  proves  the  theorem. 

This  argument  rests  upon  the  assumption  that  the  locus  of  the  equa- 
tion y  =f(^x)  is  an  unbroken  curve.  The  assumption  can  be  proved,  but 
the  proof  is  too  difficult  to  be  given  here. 


EXERCISES 
Find  the  real  roots  of  the  following  equations  graphically ; 

1.  a^-\-3x  +  2  =  0.  5.   x^-^4:X^  +  2x-3  =  0. 

2.  a;2-4a;+4  =  0.  6.    a^ -\- Aa^ -\-5  x-^2=  0. 

3.  a^  +  2«2_|_2a;  +  l  =  0.       7.  2  a.-^ -3  a;^- 12  «-{- 1  =  0. 

4.  a^-6a^-7x-6  =  0.       8.   a^ ^2x^-2x-S  =  0. 


'] 


174  COLLEGE  ALGEBRA 

147.  The  fundamental  theorem  of  algebra.  —  Every  equa- 
tion of  the  form  f(x)  =  0  has  at  least  one  root. 

This  theorem,  which  is  usually  referred  to  as  the  fundamen- 
tal theorem  of  algebra,  was  first  proved  by  Karl  Friedrich 
Gauss,  a  German  mathematician,  in  1799.  The  proof  is  too 
difficult  to  be  given  here.* 

"^N  r  148.  Theorem.  — Any  polyrtomial  f(x)  of  degree  n  has  linear 
factors  of  the  form  x  —  r,  where  r  is  a  real  or  a  complex  number. 

By  the  fundamental  theorem  the  equation 

f(x)  =  0 

has  at  least  one  root,  say  r-^.  Then  by  the  factor  theorem  f(x) 
is  divisible  by  x—r^,  and  if  /(x)  =  »(,«'' -h a^a;'*-^ -{-  ...  -f  a„,  we 
have, 

a^-  +  a,x--^-{-  ■"  +a,  =  (x-r,)(a,x--'+b,x--'+  ...  +&„_i). 
Again  by  the  fundamental  theorem  the  equation 
a.x^-''  +  b^x--'  -f-  -  +  &n-i  =  0 

has  at  least  one  root,  say  rg. 

Then  using  the  factor  theorem  again  we  see  that 

a^x^-^-j-b,x^-'-\-  ...  -{-b„_,=(x-  r,)(a,x^-'-\-  c,x--'-\-  ...  4-c„_2), 

and  that  therefore 

f(x)  =  {x  -  r^)(x  -  r,)(a,x--'--^  c,x--'-{-  ...  +  c,.^). 

By  continuing  this  argument  we  see  that  we  shall  finally  get 
f(x)  expressed  as  the  product  of  n  linear  factors.     Thus 

f(x)  =  ao(a;  -  r;)(x  -  r^)  .••  (x  -  r^).  (1) 

This  shows  that  rj,  rg,  .••,  r„  are  roots  of  the  equation 

♦  The  student  who  is  interested  will  find  a  proof  in  Fine's  College  Algebra, 
p.  588. 


THEORY  OF  EQUATIONS  175 

These  numbers  may  not  all  be  distinct.     If 

f{x)  =  a^{x  -  ViY^ix  -  ra) "2 ...  (x  -  r^)«*, 

we  agree  to  say  that  ?\  is  a  root  of  multiplicity  n^  and  to  count 
it  Ui  times.  Thus  rj  is  a  root  of  multiplicity  rii  and  we  count 
it  ?ii  times ;  rg  is  a  root  of  multiplicity  Wg  and  we  count  it  n.^ 
times ;  and  so  on.     We  have  then  the  following 

Corollary.  — Every  equation  of  degree  n  has  at  least  n  roots. 

149.  Theorem.  —  No  equation  of  degree  n  has  more  than  n 
distinct  roots. 

For  suppose  that  the  equation  is 

aoic''+  aiOf'^  4-  agic^'^H-  ...  -\-a^  =  0,  where  ao  ^  ^^ 

The  preceding  theorem  shows  that  the  left  member  can  be 
written  in  the  form 

«o(aJ  - r{)(x -  rg)  ..•  (« - O. 

If  now  r  were  a  root  of  this  equation  and  distinct  from  r^ 
r^i  '"}  r„,  we  should  have 

«oO'  -  ri)(r  -  r^)  "•  {r  -  r„)  =  0. 

But  the  first  factor,  ao,  of  the  left  member  of  this  relation  is 
by  hypothesis  not  zero.  Moreover  none  of  the  other  factors  is 
zero,  since,  by  hypothesis  r  is  different  from  rj,  rg,  •-,  r„.  But 
no  product  can  be  zero  unless  at  least  one  of  the  factors  is  zero 
(10,  Chapter  I).  Hence  the  supposition  that  r  can  be  a  root 
of /(a?)  =  0  is  wrong,  and  the  theorem  is  proved. 

Corollary.  —  Every  equation  of  degree  n  has  exactly  n  roots  in 
the  sense  explained  in  §  148. 

Corollary.  —  If  two  polynomials  in  one  variable  of  degrees  not 
greater  than  n  are  equal  to  each  other  for  more  than  n  values  of 
the  variable  the  coefficients  of  like  powers  of  the  variable  in  the 
two  polynomials  are  equal,  and  conversely. 

If        aoa;"+  aj^c^-^  +  -  +  a„=M"+  ^la:""'  +  .-.  +  b^ 


176  COLLEGE  ALGEBRA 

for  more  than  n  values  of  x,  then  the  relation 

is  satisfied  by  more  than  n  values  of  x.  If  some  of  the  coeffi- 
cients besides  the  last  one  in  the  left  member  of  this  relation 
were  not  zero,  we  should  have  an  equation  of  degree  not  greater 
than  n  with  more  than  n  roots.  But  the  theorem  shows  that 
this  is  impossible.  Hence  all  these  coefficients  are  zero,  and 
therefore  the  last  coefficient  is  also  zero.     That  is, 

a^-h^  =  0,     or     a^^b^; 
ai  —  bi  =  0,     or     a^  =  &i ; 


a^  — &„  =  0,     or     a^^  =  b^. 
The  converse  is  obvious. 

150.  Relations  between  roots  and  coefficients.  —  If  r^,  7\,  •••, 

r„  are  the  roots  of  the  equation 

x'>'  + aix'^-'^  +  ■•' +an-iX  + an  =  0,     (ao  =  1) 
then  (§  148) 

ic"  +  aix"-i  +  •••  +  ttn-ix  +  an=  (ix  —  ri)(x  -  r^)  •••  (x—  Vn). 

If  we  perform  the  indicated  multiplications  in  the  right 
member  of  this  identity,  we  shall  get  a  polynomial  which  is 
equal  to  the  left  member  of  the  identity  for  all  values  of  x. 
Hence  the  coefficients  of  corresponding  powers  of  x  in  the  two 
polynomials  must  be  equal  (§  149),  and  we  have 

ai=-ri- r2- ••• -r„.  (1) 

«2  =  riVi  +  nn  H +  nvn  +  r2r3  H h  r„_ir„.  (2) 

as  =  —  rir2r3  -  nr^n  —  •  •  •  -  r„_2r„_irn.  (3) 


a„  =  (- l)«rir2  •••»•„.  (n) 

The  second  member  of  (1)  is  the  sum  of  all  the  roots  with 
their  signs  changed.  The  second  member  of  (2)  is  the  sum 
of  the  products  of  the  roots  with  their  signs  changed   taken 


THEORY  OF  EQUATIONS  177 

two  at  a  time.  The  second  member  of  (3)  is  the  sum  of  the 
products  of  the  roots  with  their  signs  changed  taken  three  at 
a  time.     And  so  on. 

It  should  be  noted  that  these  relations  hold  only  when  the 
coefficient  of  the  highest  power  of  x  is  1.  If  it  is  not  1,  we 
must  divide  each  member  of  the  equation  by  it  before  apply- 
ing these  relations. 

^  f  (.^  151.  Imaginary  roots.  —  Although  we  are  interested  here 
only  in  the  real  roots  of  an  equation,  the  following  theorem 
concerning  the  appearance  of  imaginary  roots  will  be  useful. 

Theorem. — If  c  +  id  is  a  root  of  an  equation  with  real  coeffi- 
cients, c  —  id  is  also  a  root. 

Let  the  equation  be 

a^x^  4-  aia;"-^  H h  a„=  0, 

where  a^,  a^,  •••,  o„  are  real  numbers,  and  let  f(x)  be  a  symbol 
for  the  left  member  of  this  equation. 

The  product  of  the  two  factors  x—{g  +  id)  and  x  —  (c  —  id) 
is  x^  —  2cx  +  c^  -\-  d^.  If  now  we  divide  f(x)  by  this  product, 
we  shall  get  a  quotient  which  we  can  represent  by  the  symbol 
Q{x),  and  a  remainder  which  must  be  of  degree  less  than  2. 

^^^^  f{^)  =  Q(^)  («'  -  2ca?  +  c2  4-  d"")  4-  ra  +  r ', 

where  r  and  /  are  certain  real  constants."* 

Now  this  relation  holds  for  all  values  of  x,  and  therefore  for 
x  —  c-\-  id.     But  by  hypothesis  /(c  +  id)  =  0.     Therefore 

0=Q{c-hid){G'+2icd-d^-2<f-2icd  +  c''  +  d:')  +  r{c+id)-\-r', 
or     0  =  Q{c  +  id)  •  0  +  re  4-  ird  4-  /. 
Hence,  ^   -  re  4-/4- ?Vd  =  0. 

It  follows  from  this  by  §  123,  that  7-d  =  0  and  rc  +  r^  =  0. 
Hence  either  r  =  0  or  d  =  0. 

*  If  the  coefficients  in  f(x)  were  not  all  real,  we  could  not  be  sure  that  r 
and  r'  would  be  real. 


178  COLLEGE  ALGEBRA 

But  if  c?  =  0  the  root  c  +  id  would  be  real.  Therefore  r  =  0, 
and  then  r'  =  0. 

Hence,  /  (x)  ~  Q{x)  (x^  -  2  ex -\- c" -\-  cP), 

and  therefore 

/(c  -  id)  =  Q{c  -  ^•d)(c2-  2  icd  -  d^  -  2  c^  +  2  icd  +  c^  +d^) 

=:Q(g- id).  0  =  0. 
This  means  that  c  —  id  is  a  root  of  the  equation /(a;)  =  0. 

Corollary.  —  Every  equation  of  odd  degree  with  real  coefficients 
\\   has  at  least  one  real  root. 

152.  Transformations  of  equations.  —  In  solving  an  equation 
f(x)  =  0  it  is  frequently  necessary  to  transform  it  into  another 
equation  whose  roots  bear  a  given  relation  to  the  roots  of 
f(x)  =  0.  In  this  book  we  shall  make  use  of  three  trans- 
formations. 

153.  I.  To  traiisform  an  equatioyi  f(x)  =  0  into  one  whose 
roots  are  those  of  f(x)  =  0  with  their  signs  changed. 

If  we  write /(a.)  in  the  form  ^^^   |.^  ^  t^\\) 

f(x)  =  a^^x  -  ri)i(a;  -  r2)\-  {x  -  rj^, 
we  see  that 

/(—  a;)=  ofo(—  £c  —  ri)"i(—  x  —  ?-o)'*2 ...  (^—x  —  r^)% 

and  that  therefore  the  roots  of /(—  a?)  =  0  are—  ri,  —  rg,  — ,  —  r^. 

Moreover  each  of  these  roots  has  the  same  multiplicity  as 
the  corresponding  root  of  f{x)  =  0. 

Hence  every  root  of  f{x)  =  0  with  its  sign  changed  is  a  root 
of/(-a^)  =  0. 

If  the  given  equation  is 

a^x^  +  aio;"-^  H h  a„  =  0, 

the  transformed  equation  is 

<-^y  +  «!(-«'•)""'  +  -  +  ««=  0. 


THEORY  OF  EQUATIONS  179 

This  can  be  simplified  into 

Whether  the  sign  before  a„  is  +  or  —  depends  upon  whether 
n  is  even  or  odd.  Hence  we  have  written  this  term  (—!)'•«„, 
since  this  equals  a„  when  n  is  even  and  —  a„  when  n  is  odd. 

We  have  therefore  the  following 

Rule.  —  In  order  to  form  an  equation  whose  roots  are  those  of 
f{x)  =0  of  degree  n,  with  their  signs  changed,  change  the  signs 
of  the  alternate  coefficients  of  f(x)  =  0,  beginning  with  the  coef- 
ficient ofx"~^.  If  any  poicer  of  x  is  lacking,  its  coefficient  rmist 
he  considered  as  zero. 

Example.  —  Form  an  equation  whose  roots  are  those  of 

2x*  +  5x3-6a;2+3  =  0 
with  their  signs  changed. 

The  required  equation  is  2x^  —  5x^  —  6x^  +  ii=0. 

154.  II.  To  transform  an  equation  f(x)  =  0  into  one  whose 
roots  are  those  off(x)  =  0  each  multiplied  by  a  constant  m. 

If  we  write  f(x)  in  the  form 

f{x)  =  a^ix  -  ri)\x  -  r2)\"-  {x  -  r  JV 

we  see  that 


\mj         \m        J  \m        J 


r 


:)'\ 


,         .m 
and  that  therefore  the  roots  of 

are  mi'i,  mr^,  —,  mr^.. 

Moreover  each  of  these  roots  has  the  same  multiplicity  as 
the  corresponding  root  oif{x)  =  0. 

Hence  the  product  of  m  and  any  root  of  f{x)  =  0  is  a  root  of 


180  COLLEGE  ALGEBRA 

If  the  given  equation  is 

the  transformed  equation  is 

ao(-\+aJ^T\aJ^X~\  ...  +a.=  0. 
\7nJ  \7nJ  \7nJ 

When  cleared  of  fractions  this  becomes 

a^a;"  +  maiX'"'^  +  m^a^^-'^  +  ••.  +  w"rt„  =  0. 

We  have  therefore  the  following 

Rule.  —  In  order  to  form  an  equatiori  ^vhose  roots  are  those 
of  the  equation  f(x)=  0  of  degree  n,  each  multiplied  by  the  con- 
stant m,  midtiply  the  successive  coefficients  beginning  with  the 
coefficient  of  a;**"^  by  m,  m^,...,  ?/i"  respectively.  If  any  power 
ofx  is  lacking,  its  coefficient  must  be  considered  as  zero. 

When  m  =  —1  this  transformation  is  the  same  as  Transformation  I. 
Example  I.  —  Form  an  equation  whose  roots  are  the  roots  of 

each  multiplied  by  3. 

The  required  equation  is 

2a:5  +  32  .  5a;3  -  33  .  6a:2  +  35  .  3  =  0, 
or  2x5  +  45x3-162x2  +  729  =  0. 

Example  II.  —  Form  an  equation  whose  roots  are  the  roots  of 

jc*-3x2-6x+l  =0, 
each  divided  by  2. 

Here  m  =  \.     Hence  the  new  equation  is 

X4  _  (^)2  .  3  X2  -  (i)8  .  5  X  +Q)4  =  0, 

or  16  x*  -  12  x2  -  10  X  +  1  =  0. 

Example  IIL — Transform  the  equation 

2x8-5x2-x  +  3=:0 

into  one  whose  roots  are  the  roots  of  this  equation  each  multiplied  by  m. 
Then  find  the  least  positive  value  of  m  for  which  the  new  equation  shall 


V*^ 


THEORY  OF  EQUATIONS  181 

have  its  coefficients  all  integers  when  the  coefficient  of  the  highest  power 
of  X  is  1 . 

If  we  make  the  coefficient  of  oc^  in  the  given  equation  equal  to  1,  we  get 


Now  a:3-^x2-?^^+^^  =  0 

2  2  2. 

is  the  equation  whose  roots  are  the  roots  of  this  given  equation  each 
multiplied  by  m. 

We  wish  to  give  to  m  the  least  positive  value  that  will  make  the  coef- 
ficients of  this  last  equation  integers.  We  see  by  inspection  that  this 
value  is  2.     The  resulting  equation  is 

a:3  _  5  3-2  _  2  ic  +  12  =  0. 

Its  roots  are  equal  to  those  of  the  original  equation  each  multiplied  by  2. 

The  least  positive  value  of  m  called  for  in  problems  like  this  one  is 
either  the  coefficient  of  the  highest  power  of  x  in  the  original  equation  or 
a  divisor  of  this  coefficient. 

This  example  illustrates  an  application  of  Transformation  II  that  we 
shall  have  frequent  occasion  to  make. 

155.  III.  To  transform  an  equation  f(x)  —  0  into  one  whose 
roots  are  tJiose  off(x)=0  each  diminished  by  a  constant  h. 

If  we  write /(a?)  in  the  form 

f{x)  =  a,{x  -  ri)t<aJ  -  r^)^  -  (x  -  r J\ 

we  can  see  that 

f(x  +  h)  =  ao{x  +  h  —  ri)"i(a;  +  h  —  r^Y^  -  {x-\-h—  r^)"* 

and  that  therefore  the  roots  of 

/(x- 4-/0  =  0 
are  ?'i  —  h,  r^  —  /i,  — ,  r^^  —  h. 

Moreover  each  of  these  roots  has  the  same  multiplicity  as  the 
corresponding  root  of  f{x)  =  0.     Hence  any  root  of  f{x)  =  0 
diminished  by  ^  is  a  root  of  f(x  +  h)  =  0. 
If  the  given  equation  is 

a,x-  +  a,x--^  +  a.^""-'  +  -  +  «„=  0,  (1) 


182  COLLEGE  ALGEBRA 

the  transformed  equation  is 

a,(x  +  hy  -f  ai(x  4-  hy--"  +  a^ix  +  hf-'  +  ...  +  a,=  0.      (2) 

This  can  be  simplified  by  expanding  the  powers  of  the  bino- 
mials and  collecting  the  resulting  terras.  The  final  equation 
will  be  in  the  form 

A,x-  4-  A,x--^  +  A.x'^-'  +  -  +  A,,=  0.  (3) 

In  most  cases  the  labor  involved  in  determining  the  values 
of  Aq,  Ai,  A2,  ..♦,  An  in  this  way  is  very  great,  and  we  there- 
fore look  for  a  shorter  way  of  doing  this. 

If  in  equation  (2)  we  put  x  —  7i  in  place  of  x,  we  get  back  to 
equation  (1).  Hence,  since  the  left  member  of  (3)  is  merely 
another  form  of  the  left  member  of  (2), 

ao<K"  -f-  a^x""-^  +  a^''~^  +  -  +  a„ 

=  A,(x  -  hy+  A,{x  -  hy-'  -h  -  A,,_,(x  -  /i)  +  A 

=  [A,{x  -  hy-'-\-  A,(x  -  hy-'^+  ...  +  An_{\(x  -  h) + A. 

This  shows  that  if  we  divide  the  left  member  of  (1)  by  a;  —  /i, 
the  remainder  is  A-  If  we  divide  the  quotient  by  £c  —  7i,  the 
remainder  is  A-u  and  if  we  divide  this  second  quotient  by 
X—  h,  the  remainder  is  ^„_2.  By  working  back  in  this  way  we 
can  get  all  the  coefficients  in  the  left  member  of  (3),  A  l^eing 
the  last  quotient,  and  Ai  the  last  remainder.  We  have,  there- 
fore, the  following 

Rule.  —  In  order  to  form  the  equation  whose  roots  are  the  roots 
^ffi'x)  —  ^  ^^^^  diminished  by  hj  divide  f(x)  and  each  successive 
resulting  quotient  by  x  —  h  until  a  constaiit  quotient  is  obtained. 

Tlie  first  remainder  is  the  constant  term  and  each  successive  re- 
mainder is  the  coefficient  of  the  next  higher  power  of  x,  in  the 
desired  equation,  the  last  quotient  being  the  coefficieyit  of  the  high- 
est power  of  X. 

The  advantage  of  computing  these  coefficients  in  this  way 
lies  in  the  fact  that  the  work  can  be  shortened  by  the  use  of 
synthetic  division. 


THEORY  OF  EQUATIONS 


183 


Example.  —  Form  an  equation  whose  roots  are  the  roots  of 
2x4_5a;3_|_7a;_4  =  0 
2-    5+   0  + 


each  diminished  by  2. 


5  + 
4- 


7   -4|^ 
4  +6 


2_    1_    2  +    3 
+    4+6+8 

+  2 

2+    3+    4 
+    4  +  14 

+  11 

2+    7+18 

+    4 

2  +  11 

The  desired  equation  is 

2  a^  +  11  x3  +  18  a:2  +  11  a;  +  2  =  0. 

156.   Graphical  interpretation  of  Transformation  III.  —  If  we 

represent  the  left  member  of  equation  (3)  of  the  preceding 
article  by  F(x),  the  value  of  y  in  the  equation  y  =  F(x)  is  the 
same  f or  a;  =  a  —  7i  as  it  is  in  the  equation  y  =f(x)  for  x  =  a. 
Hence  y  =  F(x)  is  the  equation  of  the  locus  of  y  =f(x)  when  the 
y-axis  is  moved  h  units  to  the  right  of  its  original  position. 
It  is  to  be  understood  that  if  h  is  negative,  the  newiy-axis  is  to 
the  left  of  the  old  one. 

For  example,  to  move  this  axis  —  2  units  to  the  right  is  to  move  it  2 
units  to  the  left. 

Consider  the  equation  y^  —  x^  —  x  —  2  =  0, 
and  form  the  equation  whose  roots  are  the 
roots  of  this,  each  diminished  by  1. 

1_1_1_2L1_ 
+1+0-1 


1+0-1 

+  1  +  1 


1  +  1 

+  1 


+  0 


1  +  2 

The  new  equation  is  cc^  _|_  2  x^ 
The  locus  of  the  equation 

y  =  x^~x^-x-2 


3  =  0. 


ffe 


184  COLLEGE  ALGEBRA 

is  given  in  the  figure  and  the  locus  of  the  equation 

is  the  same  curve  provided  we  move  the  y-axis  one  unit  to  the  right  of  its 
original  position.    The  broken  line  is  the  new  y-axis. 


^ 


EXERCISES 


1.  Does  the  proof  of  the  second  corollary  of  the  theorem  of 
§  149  apply  when  the  two  polynomials  are  not  of  the  same 
degree  ? 

Form  equations  whose  roots  are  the  roots  of  the  following 
equations  with  their  signs  changed  : 

2.  i^-4:X^-5x-]-l  =  0,  8.  x^-{-l==0. 

3.  6a^+7a^+2fl;--a;-8=0.  9.  Sx^-2x'-{-10x'+5x+7==0. 

4.  12a;4_j_lla;2 4-2  =  0.  10.  -U^ -5x^-3x-\-2  =  0. 

5.  2x*  +  Sx^-4:X-{-l  =  0.  11.  5a^-8x~-x-^2  =  0, 

6.  a^  =  3x^-2x-\-5.  12.  3x^-\-7 c(^-{-5x^+2x-\-S=0, 

7.  x^-x*-\-a^-x^-\-x-l  =  0,  13.  -2a;^-3a7^-fl  =  0. 

Form  equations  whose  roots  are  the  roots  of  the  following 
equations  each  multiplied  by  the  numbers  placed  opposite  the 
respective  equations : 

14.  aj*  +  2a^  +  3a^  +  4a;-f-5  =  0,     2. 

15.  5.^-a^  +  4  =  0,     3. 

16.  2x^  +  10x^-7x^  +  x-\-4:  =  0,     i 

17.  5a;^-a^+7a^+3x-4  =  0,     5. 

18.  a3  +  a;2  +  a;  +  l  =  0,     -2. 

19.  -3x'-^10x^  +  6x'-x-4:  =  0,    |. 

Form  equations  whose  roots  are  the  roots  of  the  following 
equations  each  multiplied  by  m.     Then  find  the  least  positive 


THEORY  OF  EQUATIONS  185 

value  of  m  for  which  the  coefficients  of  the  resulting  equation 
in  each  case  are  integers  when  the  coefficient  of  the  highest 
power  of  a;  is  1. 

20.  6a^4-5a^  +  a;-2  =  0.  25.  5a^  +  2a;  +  l  =  0. 

21.  aj^-2a^  +  7ar'  +  -+3=0.     26.  6^ +  2^ +  x-2>  =  0. 
22.20.^  +  40.  =  !.       ^  21.  2x^-1  ;^-Wx^+x-S  =  0. 

28.    Za^+x'^-2y^-4.x'-Qx-2 
23.  a.'3+2V  =  0.  ^  ^^ 


24.  9a.*-4x^-6a.24-9a.  +  18     29.  3a;*4-a^  +  a;  +  5  =  0. 

=  0.   30.   -2a.3_|_4^^^6a.  +  8  =  0. 

31.  What  are  the  roots  of  ar^  +  2ar  — 2  a. +  3  =  0,  it  being 
given  that  3  is  a  root  of  ar'  —  2  a?-  —  2  a;  —  3  =  0  ? 

32.  What  are  the  roots  of  ar^  +  4  a.^  —  7  a;  —  10  =  0,  it  being 
given  that  —  2  is  a  root  of  a;^  —  4 a;-  —  7 a; -|-  10=  0  ? 

Form  equations  whose  roots  are  the  roots  of  the  following 
equations,  each  diminished  by  the  number  placed  opposite 
the  respective  equation.  Then  plot  the  loci  of  the  equations 
obtained  by  putting  the  left  members  of  the  original  equa- 
tions equal  to  y,  and  show  the  effect  on  the  ?/-axis  of  each 
transformation. 

33.  a.2_53._^g^Q^     I  ZS.  x^-.^x  +  Q  =  0,     -1. 

34.  ar^_3a.-2-a.  +  3  =  0,     2.  39.  x2  +  3a;  +  2  =  0,     2. 

35.  a.3-l  =  0,     1.  40.  a.2-2a;  +  l  =  0,     1. 

36.  a,-3  +  6a.2+9a;  +  20=0,     3.  41.  x^- 3a;-  +  3a.-l  =  0,     1. 

37.  a.^  +  6a.  =  8  +  ar^     2.  42.  ar^- Gaj^-lla;  -  6  =  0,     2. 

43.  What  relations  do  the  loci  of  the  equations  y  =  —f(x) 
and  y=f(—x)  bear  to  the  locus  of  y=f(x)?  Apply  your 
answer  to  the  cases  in  which  f(x)  is  each  of  the  polynomials 
in  Exs.  33-37  in  turn.       ^w   -       :^'<t^ 


\n 


186  COLLEGE  ALGEBRA 

157.  Continuations  and  variations  in  sign.  —  The  presence  of 
two  consecutive  terms  with  like  signs  in  a  polynomial  with  real 
coefficients  is  called  a  continuation  in  sign,  and  the  presence  of 
two  consecutive  terms  with  unlike  signs  is  called  a  variation  in 
sign. 

Thus,  inSx^  —  7  x'^  —  x^  -\-x-{-S  there  are  two  continuations  and  two 
variations  in  sign.  It  is  not  necessary  to  take  into  account  the  missing 
terms. 

Consider  the  product  arising  from  the  multiplication  of 

3  a;4  -  2  aj3  4. 5  a^~4.~  3  a;  _^  2  " 
x-2 

3a;^-2aj4  +  5a^+    3x''-^2x 

—  6af  +  4:X^-10x'^-6x-4: 
3x^-Hx^-{-9x^-  7  x"- 4.x -4. 

Here  there  are  two  variations  in  sign  in  the  original  poly- 
nomial, and  three  variations  in  the  product.  This  illustrates 
what  happens  in  general,  as  is  shown  by  the  following 

Theorem.  —  If  f(x)  is  a  'polynomial  with  real  coefficients  and 
r  is  a  positive  number,  the  number  of  variations  in  sign  in 
(x  —  r)f{x)  is  at  least  one  more  than  the  number  of  variations  in 

The  proof  of  this  theorem  is  based  on  the  following 

Underlying  principle.  —  If  in  a  sequence  of  real  numbers 
Cij  C2,  •  •  *,  Cp,  the  signs  of  Cj  and  c^  are  opposite,  the  sequence  has  at 
least  one  variation  of  sign. 

If  f{x)  =  a,x-  +  a,a^-'  +  -  +  a,_,a^, 

where  a^  and  a^-^  ^^^  different  from  0,  and  k  is  greater  than,  or 
equal  to,  0,  then 

(x-r)f(x)  =  A,x^+'+A,x^+  ...  +A-.a^-''+A-.+i«'-*, 
where  A^  =  «„,  ^„_,+i  =  -  ra,_„ 


THEORY  OF  EQUATIONS  187 

and  for  all  other  coefficients, 

In  this  last  formula  p  can  have  any  value  from  1  to  n  —  Jc 
inclusive. 

If  any  two  consecutive  coefficients  of  f(x),  as  a^.j  and  a^, 
have  opposite  signs,  A^  has  the  same  sign  as  a^.  For  the  for- 
mula for  Ap  shows  that  it  is  positive  when  ap_i  is  negative  and 
ttp  is  positive,  and  that  it  is  negative  when  a^.j  is  positive  and 
cip  is  negative. 

Suppose  now  that  the  last  variation  in  sign  in  the  sequence 
Qq,  «!,  —,  a^^j,  occurs  between  o,_i  and  aj.  Then  since  A^  =  a^ 
there  must  be  at  least  as  many  variations  in  sign  in  the  se- 
quence Aq,  Ai,  •••,  Ai  as  there  are  in  the  sequence  a^,  a^,  — ,  aj,  and 
therefore  at  least  as  many  as  there  are  in  the  sequence  a^y  a„ 
•••)  <^n-k-  I^i^t  there  must  be  at  least  one  variation  in  sign  from 
Ai  to  A^-k^-i  since  the  former  has  the  same  sign  as  a,,.;^  and  the 
latter  has  the  opposite  sign  (An_k+i  =  —  ^'<*n-t)- 

Hence  there  is  at  least  one  more  variation  in  sign  in 
{x  —  r)f{x)  than  inf(x).  " 

158.   Descartes's  Rule  of  Signs.  —  An  equation  f(x)=  0  with 
real  coefficients  cannot  have  more  positive  roots  than  f(x)  has  vor 
riations  in  sign,  nor  more  negative  roots  tf^an  f(—x)  has^varior     . 
tions  in  sign.M,"^'J^  r^ ,  J  4^  yj-j  (^fV^^-T^-*}  co  -i^  <i^^  vf 

I.    Suppose  that  JlfU^-^  V^Ckaa,    A^im-a^   -rVO^t-^cA^^ 
ri,  r2,  •••,  Vp  are  the  positive  roots  otf{x)=  0, 
and  that  f(x)  =  (x—  ri){x  —  r^)  •••  {x  —  rp)Q(x). 

The  number  of  variations  in  sign  in  (x  —  r^)  Q(x)  exceeds  the 
number  of  variations  in  Q(x)  by  at  least  one.  And  similarly 
when  we  multiply  by  each  of  the  p  —  1  remaining  factors 
(x  —  r^),  (x  —  7'2),  •••,  (x  —  rp_i)  in  turn  we  add  at  least  one  va- 
riation in  sign  to  the  product.  Hence  there  must  be  at  least  p 
variations  in  sign  in  f{x). 

k 


188  COLLEGE  ALGEBRA 

II.  Since  the  negative  roots  of  f(x)=0  are  the  positive 
roots  of /(— 0?)=  0  (§  153),  the  number  of  these  negative  roots 
cannot  exceed  the  number  of  variations  in  sign  in/(—  x). 

It  should  be  observed  that  Descartes's  rule  of  signs  does 

not  tell  us  how  many  positive  or  negative  roots  the  equation 

f{x)=  0  has;  it  merely  tells  us  that  the  equation  cannot  have 

more  than  a  certain  number  of  each  of  these  kinds  of  roots. 

It  gives  us  no  information  about  the  number  of  zero  roots. 

But  these  roots  can  readily  be  determined  by  inspection.     In 

finding  the  ot*her  real  roots  it  is  best  first  to  divide /(cc)  by  the 

highest  power  of  x  by  which  it  is  divisible.     If  this  power  is 

the  ath,  and  ^,  .        „.  /  ^ 

f(x)=x%(x), 

then  0  is  a  root  of  f(x)  =  0  of  multiplicity  a  and  the  non-zero 
roots  of  f{x)  =  0  are  the  roots  of  /i  (x)  =  0. 

If /(»)  has  just  one  variation  in  sign,  it  can  be  shown  that 
the  equation  f{x)  =  0  has  one  positive  root.  We  shall  omit  the 
proof  of  this  statement. 

EXERCISES 

Use  Descartes's  rule  of  signs  to  obtain  all  the  information 
you  can  about  the  number  of  positive  and  of  negative  roots  of 
each  of  the  following  equations.  Can  you  tell  whether  any  of 
the  equations  have  imaginary  roots  ? 

1.  x'  +  Sx''-\-5x-l  =  0.  6.  iB^4-3«2-2ic-5  =  0. 

2.  x*-[-Sx'^-5x-\-l  =  0.  7.  4:X^-\-af-^5ij(^-\-3  =  0. 

3.  x'-Sx^-{-5x  +  l  =  0.  8.  a;«-2  =  0. 

4.  x'-6  =  0.  9.  x^i-2x^-\-Sx  +  4.  =  0. 

5.  2x^  +  7x'-^x'^+5  =  0.  10.  3x»+16a;2+18a;-20=0. 

11.    The  roots  of  the  equation 

6x'-\-29a^-54.x'-51x-10=0 

are  all  real.     How  many  of  them  are  positive  and  how  many 
are  negative  ? 


THEORY  OF  EQUATIONS  189 

12.   The  equation     6x*  —  x^-x^-75x-25  =  0 
has  an  imaginary  root.     How  many  of  its  roots  are  negative  ? 

^    13.    Show  that  the  equation 

2x''-5x--x  +  2  =  0 
has  at  least  two  imaginary  roots. 

159.   Theorem.  — Every  rational  root  of  the  equation 

a?"  4-  Oio:"-^  +  a.a;«-2  + h  «n  =  0 

with  integral  coefficients  is  an  integer  arid  a  divisor  of  a^. 

Suppose  that  the  given  equation  has  a  fractional  root,  and 
that  this  fraction  in  its  lowest  terms  is  -^'     This  means  that 

q 

q>l  and  that  jp  and  q  have  no  common  divisor  except  1. 
Then  we  have  the  relation 

qn  qu    1  gn    . 

Multiplying  both  members  by  ^""^  and  transposing,  we  get 

Q 
Now  the  right  member  of  this  relation  is  an  integer,  and 
therefore  if  ^  were  a  root  of  the  given  equation,  p"  would  be 

divisible  by  q.  But  this  is  impossible,  since  p  and  q  have  no 
common  divisor  except  1  and  q>l.  Hence  every  rational 
root  of  the  equation  must  be  an  integer. 

If  r  is  an  integral  root  of  the  equation,  we  have 

r"  +  air"-i-|-  agr^-^H 1-  a„  =  0, 

or  r""-^  +  air"-2  +  cfgr^-^  H \-  a„_i  =  -  ^ . 

r 

Now  the  left  member  of  this  relation  is  an  integer  and  there- 
fore —  is  an  integer.     This  is  equivalent  to  saying  that  a"  is 

divisible  by  r. 

This  completes  the  proof  of  the  theorem. 


190  COLLEGE  ALGEBRA 

160.  We  are  now  in  a  position  to  solve  the  main  problem  of 
this  chapter;  namely,  to  find  the  real  roots  of  an  equation 
with  rational  coefficients. 

We  consider  first  the  problem  of  finding  the  rational  roots  of 
such  an  equation. 

161.  Rational  roots.  —  If  the  equation  is  of  the  form 

ttox-  +  Oia;"-'  +  a^--'  +  . . .  4-  a„  =  0,  (1) 

where  aQ  ^  0,  we  first  form  a  new  equation  whose  roots  are  711 
times  the  roots  of  this  equation,  and  find  the  least  positive 
value  of  m  for  which  this  new  equation  has  integral  coefficients 
when  the  coefficient  of  its  highest  power  of  x  is  1.  Suppose 
that  the  new  equation  for  this  value  of  m  is 

a;"  +  b^x^-'^  +  h^""-^  H h  &„  =  0  (2) 

in  which  the  coefficients  are  integers.  If  equation  (1)  has 
a  rational  root,  equation  (2)  has  an  integral  root  which  is  a 
divisor  of  6„. 

We  therefore  determine  by  synthetic  division  what  positive 
and  negative  divisors  of  6„  are  roots  of  equation  (2).  If  we 
divide  every  such  root  by  m,  we  shall  get  all  the  rational  roots 
of  equation  (1). 

When  one  root  r  of  (2)  has  been  found,  divide  the  left  mem- 
ber of  (2),  which  we  will  call  F(x),  by  x  —  r. 

F{x)  =  (x-r)Q(x). 

Every  other  root  of  (2)  is  a  root  of  Q(x)=0,  and  these  other 
roots  can  best  be  obtained  from  this  last  equation.  We  shall 
refer  to  it  as  the  depressed  equation  since  it  is  of  lower  degree 
than  (1). 

Sometimes  the  labor  of  finding  these  roots  can  be  shortened 
by  determining  the  maximum  number  of  positive  and  negative 
roots  by  means  of  Descartes's  rule  of  signs. 


THEORY  OF  EQUATIONS  191 

Example.  —  Find  the  rational  roots  of 

9x4  +  12x3+ 10x2  +  3;- 2  =  0.  (1) 

The  equation  whose  roots  are  m  times  the  roots  of  this  can  he  written 
in  the  form  ,  ^     „ 

The  least  positive  value  of  m  that  will  make  these  coefficients  integers 
is  3.     For  this  value  of  m  the  equation  is 

x*  +  4  x3  +  10  x2  +  3  X  -  18  =  0.  (2) 

"We  know  from  §  158  that  this  equation  has  just  one  positive  root  and 
from  §  159  that  this  root  is  a  divisor  of  18,  if  it  is  rational.  Now  the 
positive  divisors  of  18  are  1,  2,  3,  6,  9,  18  ;  and  we  find  by  synthetic 
division  that  1  is  a  root. 

1+4  +  10+    3- 18  U_ 

-f- 1  4-    5  +  15  +  18 
1  +  5  +  15  +  18+    0 

All  the  other  roots  are  roots  of  the  depressed  equation 
a;3  +  5ic2  4.i5a;  +  i8  =  o. 

This  equation  has  no  positive  root,  and  any  negative  root  it  may  have  is 
a  negative  divisor  of  18. 

Try  -  1.  1  +  5  4.  15  4.  18  |  -1 

_1-    4-11 
1+4  +  11+    7 
Hence  —  1  is  not  a  root. 

Try  -  2.  1  +  5+15  +  18  [  -2 

-2-    6-18 
1  +  3+9+0 

Hence  —  2  is  a  root,  and  the  other  roots  are  roots  of  the  depressed 

and  therefore  imaginary. 

Since  1  and  —  2  are  all  the  rational  roots  of  (2) ,  i  and  —  |  are  all  the 
rational  roots  of  (1),  since  the  roots  of  (2)  are  the  roots  of  (1)  each  mul- 
tiplied by  3. 

It  is  not  necessary  to  perform  any  extra  work  to  get  the  successive  de- 
pressed equations  since  the  left  member  of  each  one  is  obtained  inciden- 
tally in  finding  a  root  of  the  preceding  equation. 


192  COLLEGE  ALGEBRA 

EXERCISES 

Find  all  the  rational  roots  of  each  of  the  following  equa- 
tions :                                        \ 

1.  a^4.a;2  4-aj4-l  =  0.          1  11.  6  a^-2x^ -\-Ax-l  =  0. 

2.  2a;3  4.a;2_^a;-l  =  0.         '  12.  4.a^-l()x'^ -9x -}-S6=0. 

3.  a:^  +  5a^-2x-\-2  =  0.  13.  4:X^ +  S  x"  -  x-2  =  0. 

4.  x^-5a^-\-10x-Tx-2  =  0.  14.  6x'  +  2a^  +  5  =  0. 

5.  a^-\.x'-16x  +  20=:0.  15.  4.x^-5x-6  =  0. 

6.  2  a.-3  +  9  3.-2  +  110;  + 3  =  0.  16.  3ar^+16  a-^+lS  a;-20=0. 

7.  x^-^Sx^  =  a^.  17.  2a;3  +  3i»2  +  5a;  +  2  =  0. 

8.  a;*  +  ic3_|_3j2_j_^_^;L^0^  ;L8.  a;^  +  3  ic2  + 2  =  0. 

9.  a;3_3a^  +  5^.  +  4  =  0.  19.  x^-3  x^-{-S  x^-3  xi-2=0. 
10.   8a;3_4a.2_2aj  +  i  =  o.         20.    a^-32  =  0. 

162.  Irrational  roots.  —  The  first  step  in  finding  an  irra- 
tional real  root  of  f(x)  =  0  is  to  find  two  consecutive  integers 
between  which  this  root  lies.  This  can  be  done  by  making  use 
of  the  theorem  of  §  146. 

The  following  method  for  computing  approximately  the  ir- 
rational real  roots  of  an  equation  is  based  upon  this  theorem. 
It  is  known  as  Horner's  Method,  from  the  name  of  its  in- 
ventor. 

163.  Positive  irrational  roots.  —  The  essential  features  of 
this  method  can  best  be  explained  by  means  of  an  example. 

Example.  —  Find  the  irrational  real  roots  of  the  equation 

0^  +  3  a;2  -  2  a?  -  5  =  0.  (1) 

1.  The  left  member,  which  we  represent  by  f(x),  has  one 
variation  of  sign  and  therefore  the  equation  has  just  one  posi- 
tive root. 

2.  In  order  to  determine  the  location  of  this  positive  root, 
we  substitute  for  x  in  f{x)  successive  integral  values  of  x  be- 


THEORY  OF  EQUATIONS  193 

ginning  with  0.     We  see  directly  that  /(O)  =  —  5  and  indirectly 
by  means  of  synthetic  division  that /(I)  =  —  3  and/(2)  =  11. 

l  +  3_2-5[l_  1+3-    2-    b\2_ 

+  1+4  +  2  +2  +  10+16 

1  +  4+2-3  1  +  5+    8  +  11 

Hence  by  the  theorem  of  §  146,  the  equation  has  a  root  be- 
tween 1  and  2. 

3.  Form  a  new  equation  whose  roots  are  equal  to  the  roots 
off(x)=0  each  diminished  by  1. 

1  +  3  -2  -5[1_ 
+  1   +4  +2 


1+4  +2 
+  1  +6 


1  +  5 

+  1 

1  +  6 


+  7 


This  new  equation  is 

0,^  +  6  0-^4- 7  a;- 3  =  0,  (2) 

and  it  has  a  root  between  0  and  1. 

4.  For  a;  =  0,  .1,  .2,  ,3,  and  .4  the  left  member  of  this  equa- 
tion equals  respectively  —3,  —2.239,  —1.352,  —.333,  and 
.824,  as  may  be  seen  from  the  following  computations : 

1  +  6+7-3         y.  1+6+7-3         [^ 

+    .1  +    .61  +    .761  +    .2  +  1.24,+  1.648 


1  +  6.1  +  7.61  -  2.239  1  +  6.2  +  8.24  -  1.352 

1+6     +7       -3       Lii  1  +  ^     +7       -3       [A^ 

+    .3  +  1.89  +  2.667  +    .4+2.56  +  3.824 

1  +  6.3  +  8.89-    .333  1  +  6.4+9.56+    .824 

Hence  equation  (2)  has  a  root  between  .3  and  .4. 
5.    Form  a  new  equation  whose  roots  are  the  roots  of  (2) 
each  diminished  by  .3. 


194  COLLEGE  ALGEBRA 

1  +  6+7       -3         |_^ 
.3  +  1.89  +  2.667 


1  +  6.3  +  8.89 
+    .3  +  1.98 

— 

.333 

1  +  6.6 
.3 

+  10.87 

1  +  6.9 

The  new  equation  is 

a?  +-  6.9  x"  +  10.87  X  -  .333  =  0  (3) 

and  it  lias  a  root  between  0  and  .1. 

6.   We  find  by  trial  that  this  root  lies  between  .03  and  .04. 

.03 


.04 


1+ 6.94  +  11.1476  +  .112904 

7.   Form  a  new  equation  whose  roots  are  the  roots  of  (3)  each 
diminished  by  .03. 

1+6.9    +  10.87      -  .333        1_^ 
.03  +      .2079  +  ..332337 


1  +  6.9    +  10.87      - 
.03  +      .2079  + 

.333 
•332337 

1  +  6.93  +  11.0779  - 

1+6.9    +  10.87      - 
+    .04+      .2776  + 

.000663 

.333 
.445904 

1  +  6.93+11.0779 
+    .03+      .2088 


-  .000663 


1  +  6.96 
+    .03 


+  11.2867 


1  +  6.99 

The  new  equation  is 

0?  +  6.99  x"  +  11.2867  x  -  .000663  =  0  (4) 

and  it  has  a  root  between  0  and  .01. 
8.  We  find  by  trial  that  this  root  lies  between  0  and  .001. 

1  +  6.99  +  11.2867      -  .000663        |  .001 

+    .001  +      .006991  +  .011293691 
1  +6.991  +  11.293691  +  .010630691 

Obviously  the  left  member  of  (4)  is  negative  for  ic  =  0. 


THEORY  OF  EQUATIONS 


195 


9.  Since  each  root  of  (4)  is  .03  less  than  a  root  of  (3),  the 
latter  has  a  root  between  .03  and  .031. 

Since  each  root  of  (3)  is  .3  less  than  a  root  of  (2),  the  latter 
has  a  root  between  .33  and  .331. 

Since  each  root  of  (2)  is  1  less  than  a  root  of  (1),  the  latter 
has  a  root  between  1.33  and  1.331. 

Hence  we  have  found  a  root  of  the  original  equation  correct 
to  two  decimal  places.  If  the  digit  in  the  third  decimal  place 
had  been  5  or  larger,  the  root  to  two  decimal  places  would 
have  been  1.34.  In  general,  in  order  to  get  a  root  to  r  decimal 
places,  it  is  necessary  to  determine  whether  the  digit  in  the 
(r  +  l)th  place  is  less  than  5,  or  not. 

In  practice  the  body  of  this  work  can  be  compactly  arranged 

as  follows : 

1  +  3        _   2  -5  \1 

+1+4  +2 


L§ 


1+4 
+  1 

+    2 
+    6 

-3 

1  +  5 

+  1 

+    7 

1  +  6 

+    .3 

+    7 
+    1.89 

-3 

+  2.667 

1  +  6.3 
+    .3 

+    8.89 
+    1.98 

-    .333 

1+6.6 
+    .3 

+  10.87 

1+6.9 

+    .03 


+  10.87       - 

+      .2079  + 


.333 

.332337 


|.03 


1  +  6.93    +  11.0779 
+    .03  .2088 


1+6.961  +  11.2867 

.03 1 
1  +  6.99    +  11.2867  - 


.000663 


.000663 


164.  By  an  obvious  continuation  of  this  method  we  can  com- 
pute this  root  to  any  number  of  decimal  places. 

The  higher  powers  of  a  number  between  0  and  1  are  less 
than  the  number  itself.  Hence  if  an  equation  is  known  to 
have  a  root  between  0  and  1,  the  terms  containing  the  higher 


\^ 


196  COLLEGE  ALGEBRA 

powers  of  the  unknown  will  be  relatively  unimportant,  and 
the  root  can  be  determined  approximately  by  neglecting 
these  higher  powers  and  solving  the  resulting  equation 
of  the  first  degree.  The  nearer  the  root  is  to  zero,  the  closer 
this  approximation  will  be.  Applying  this  principle  to  (4) 
we  see  that  .00006  is  an  approximate  value  of  one  of  its 
roots,  and  this  suggests  that  this  root  lies  between  0  and 
.001,  as  was  stated.  But  it  must  be  kept  clearly  in  mind 
that  this  is  only  a  suggestion,  and  that  the  suggestion  must 
be  tested. 

In  a  similar  way  we  get  the  suggestions  that  .03  and  .4  are 
approximate  roots  of  (3)  and  (2)  respectively. 

These  considerations  enable  us  to  do  away  with  many  syn- 
thetic divisions  that  otherwise  would  be  necessary. 

If  in  our  efforts  to  locate  a  root  between  two  consecutive  in- 
tegers we  divide /(a;)  hj  x  —  r,  where  r  is  positive,  and  find  that 
the  remainder  and  all  the  coefficients  of  the  quotient  are  posi- 
tive, we  may  conclude  that  r  is  greater  than  any  real  root  of  the 
equation  f(x)  =  0.  For  if  we  divide  f(x)  by  a?  —  rj,  where 
Vi  >  r,  all  these  coefficients  after  the  first  one,  as  well  as  the 
remainder,  will  be  greater  than  they  were  and  therefore  the 
remainder  will  not  be  zero.  The  value  of  this  observation  may 
be  seen  in  step  3  of  the  example  worked  above.  The  sugges- 
tion just  described  tells  us  that  the  root  we  are  looking  for  is 
approximately  .4,  and  we  accordingly  divide  the  left  member  of 
equation  (2)  by  a;  —  .4. 

1  +  6     +7       -3        |_^ 

+    .4  4-2.56  +  3.824 
1+6.4  +  9.56+    .824 

Since  all  these  sums  are  positive,  we  know  that  the  root 
cannot  lie  between  .4  and  .5,  and  we  accordingly  divide  by 
X  —  .3  and  note  the  sign  of  the  remainder. 

There  are  other  considerations  by  means  of  which  this 
process  can  be  shortened,  but  we  shall  not  take  them  up 
here. 


THEORY  OF  EQUATIONS  197 

When  one  irrational  root  has  been  found  to  the  required 
degree  of  approximation,  an  entirely  new  start  must  be  made 
in  finding  the  next  one.  On  the  other  hand,  when  a  rational 
root  has  been  found,  the  equation  can  be  depressed  (§  161) 
and  the  finding  of  the  remaining  real  roots,  both  rational  and 
irrational,  is  thereby  made  correspondingly  easier. 

165.  Negative  irrational  roots.  —  In  order  to  find  the  nega- 
tive irrational  roots  of  f{x)  —  0,  find  the  positive  irrational 
roots  of /(— a;)=0.  These  with  their  signs  changed  are  the 
roots  sought. 

166.  Summary.  —  In  order  to  find  all  the  real  roots  of  an 
equation  f(x)  =  0,  in  which  f(x)  is  a  polynomial  with  rational 
numerical  coefficients,  proceed  as  follows  : 

1.  Find  all  the  rational  roots  by  the  method  described  in 
§  161.  When  each  rational  root  has  been  found  depress  the  cor- 
responding  equation. 

2.  See  if  the  last  depressed  equation  after  all  the  rational 
roots  have  been  found  has  any  positive  rootSy  and  determine  by 
synthetic  division  two  consecutive  integers  between  which  such  a 
root  lies. 

3.  Form  a  new  equation  tchose  roots  are  the  roots  of  this  equa- 
tion each  diminished  by  the  smaller  of  these  tico  integers. 

4.  Tlie  residting  equation  has  a  root  between  0  and  1.  Find 
by  synthetic  division  the  two  consecutive  tenths  between  which  the 
root  lies. 

5.  Form  a  new  equation  whose  roots  are  the  roots  of  this  equa- 
tion each  diminished  by  the  smaller  of  these  tenths. 

6.  Tlie  residting  equation  has  a  root  between  0  and  .1.  Find 
by  synthetic  division  two  consecutive  hundredths  between  which 
this  root  lies. 

7.  Form  a  neiv  equation  whose  roots  are  the  roots  of  this  equa- 
tion each  diminished  by  the  smaller  of  these  hundredths. 

8.  If  the  root  is  required  to  r  decimal  places ^  continue  this  pro- 
cess until  r  -f  1  decimal  places  have  been  determined. 


198  COLLEGE  ALGEBRA 

9.  Add  up  the  amounts  by  ichich  the  roots  of  the  successive 
equations  have  been  diminished.  This  sum,  with  the  figure  in  the 
rth  decimal  place  increased  by  1  in  case  the  figure  in  the  (r  +  l)th 
place  is  5  or  more,  is  the  root  sought  to  the  required  degree  of  ap- 
proximation. 

10.  If  there  are  other  positive  irrational  roots,  find  each  of  them 
in  the  same  way. 

11.  In  order  to  find  the  negative  irrational  roots,  fiyid  the  posi- 
tive irrational  roots  off{—x)  =  0  a7id  change  the  sign  of  each  one. 

12.  Make  use  of  all  the  information  that  is  obtainable  from 
Descartes' s  rule  of  signs  as  to  the  number  of  positive  and  of  neg- 
ative roots. 

If  in  finding  the  irrational  roots  of  f{x)  =  0  by  Horner's 
Method  we  have  any  reason  to  suspect  the  existence  of  two 
roots  between  a  and  a  +  1,  we  should  examine  the  sign  of  f{x) 
for  x  —  a,  x  =  a-\-^,  and  x  =  a-[-l;  and  it  may  be  necessary  to 
examine  the  sign  of  f{x)  for  values  of  x  still  closer  together. 

EXERCISES   AND   PROBLEMS 

Find  the  values  of  the  real  roots  of  the  following  equations 
correct  to  two  decimal  places : 

1.  2a^  +  4aj2-10a;  +  3  =  0. 

2.  5a^-3a;2- 6x4-3  =  0. 

3.  x4_^a^-7a;2-8a;+20=0. 

4.  2a^-aj3-a;2-a;-3  =  0. 

5.  4  0^3 ^3  a;- 3. 

11.  How  deep  will  a  cork  sphere  4  inches  in  diameter  sink 
in  water,  the  specific  gravity  of  the  cork  being  .2  ? 

Hint.  —  The  volume  of  a  spherical  segment  of  one  base  is  given  by  the 
formula  .  o  ,  ,      q 

where  x  is  the  altitude  of  the  segment  and  ri  is  the  radius  of  its  base. 


6. 

2a;3_3^2^^^0. 

7. 

8a;4  +  8ic2_  1^  =  0. 

8. 

8a;4  =  8a;2_3. 

9. 

x^-2x-2=^0. 

10. 

a^-4a;-2  =  0. 

THEORY  OF  EQUATIONS  199 

12.  How  deep  will  a  sphere  of  pine  of  specific  gravity  i  sink 
in  water? 

13.  If  the  specific  gravity  of  ice  is  .9,  how  much  of  a  sphere 
of  ice  2  feet  in  diameter  would  protrude  above  the  water  in 
which  the  ice  is  floating  ? 

14.  The  volume  of  a  box  10  x  12  x  15  inches  is  to  be  in- 
creased 50  cubic  inches  by  adding  the  same  amount  to  each  di- 
mension.    What  should  this  amount  be  ? 

15.  How  thick  should  a  hollow  spherical  shell  be  whose  in- 
ner radius  is  3  inches  in  order  to  contain  40  cubic  inches  ? 

16.  If  a  is  the  cosine  of  an  angle  and  x  is  the  cosine  of  one 
third  of  this  angle,  then     4  a^  =  3  a;  +  a. 

What  is  the  cosine  of  an  angle  of  20°  ? 
Hint.  —The  cosine  of  60°  is  |. 

17.  A  house  may  be  bought  for  $3400  cash,  or  in  annual  in- 
stallments of  $1000  each,  payable  1,  2,  3,  and  4  years  from 
date.     What  is  the  annual  interest  rate  implied  in  this  offer  ? 

Hint.  —  The  amount  of  i$3400  for  4  years  should  equal  the  sum  of  the 
amount  of  the  first  payment  for  3  years,  the  amount  of  the  second  pay- 
ment for  2  years,  the  amount  of  the  third  one  for  1  year,  and  the  fourth 
payment. 

Hence,  if  x  is  the  rate  of  interest, 

3400(1  4-  xy  -  1000(1  +  xy  -  1000(1  +  x)-^  -  1000(1  +x)-  1000  =  0, 
or,  17(l  +  x)4-5(l  +  x)3-6(l4-a:)2-5(H-a;)  -5  =  0. 

18.  A  house  may  be  bought  for  $2800  cash,  or  in  annual 
installments  of  $1000  each  payable  1,  2,  and  3  years  from  date. 
What  is  the  annual  rate  of  interest  implied  in  this  offer  ? 

19.  An  open  box  is  to  be  made  from  a  rectangular  piece  of 
tin  18  inches  long  and  10  inches  wide,  by  cutting  out  equal 
squares  from  the  corners  and  turning  up  the  sides.  How  large 
should  these  squares  be  in  order  that  the  box  contain  168  cubic 
inches  ? 


200  COLLEGE  ALGEBRA 

20.  Find  the  cube  root  of  3. 

Hint.  —  Find  the  approximate  value  of  the  real  root  of  the  equation 

21.  Find  the  cube  root  of  — 17. 

22.  Find  the  fifth  root  of  10. 

23.  Find  the  cube  root  of  115. 

Find  the  points  of  intersection  of  the  following  curves : 

24.  a;2 4-2/2  =  10,        25.    x^-^y^  =  9,       26.    4:x'^  +  7f  =  16, 

y  =  x'^  +  x-\-l.  y  —  x'^  —  x.  x  +  5y  =  y'^ -{-2. 


CHAPTER   XIV 


DETERMINANTS 


167.  In  Chapter  V  we  used  determinants  of  the  second  and 
third  orders  to  advantage  in  the  solution  of  systems  of  linear 
equations  in  two  and  three  unknowns  respectively.  Analogous 
symbols,  which  are  called  determinants  of  order  n  can  be  used 
in  the  solution  of  systems  of  n  linear  equations  in  n  unknowns, 
where  n  is  any  positive  integer.  For  values  of  n  greater  than 
3  there  is  a  greater  advantage  in  this  use  of  determinants  than 
there  is  in  the  simple  cases  in  which  n=  1,  2,  or  3. 


The  symbol 


order  and  by   definition 


was  called  a  determinant  of  the  second 


=  tti^a  —  a^i ;    likewise,   by 


definition  of  a  determinant  of  the  third  order, 


We  wish  to  define  a  determinant  of  order  w,  where  n  is  any 
positive  integer,  in  a  way  that  will  be  consistent  with  these 
definitions  of  determinants  of  order  two  and  three  respectively. 

168.  Inversions  of  numbers.  —  But  we  must  first  introduce 
the  notion  of  an  inversion.  We  use  this  term  to  describe  the 
presence  in  an  arrangement  of  positive  integers  of  a  greater 
integer  before  a  smaller  one. 

Thus,  in  the  arrangement  1,  2,  4,  5,  6,  3,  the  integers  4,  5,  and  6  each 
appear  before  the  smaller  one  3,  and  there  are  in  this  arrangement  there- 
fore three  inversions.  There  are  no  inversions  in  the  arrangement  1,  2, 
3,  4,  5,  6 ;  and  there  are  two  in  the  arrangement  2,  1,  3,  5,  4,  6. 

201 


202  COLLEGE  ALGEBRA 

Theorem.  —  If  in  an  arrangement  of  positive  integers  a7iy  two 
of  the  integers  he  iyiterchanged,  the  number  of  inversions  is  in- 
creased or  diminished  by  an  odd  number. 

Consider  first  the  effect  of  interchanging  two  adjacent  inte- 
gers a  and  b.  Every  integer  that  preceded  both  a  and  h  in  the 
original  arrangement  will  precede  them  in  the  new  arrangement, 
and  every  integer  that  followed  them  originally  will  follow 
them  in  the  new  arrangement.  Moreover,  the  only  relative 
positions  that  are  changed  are  those  of  a  and  6,  and  the  effect 
of  this  change  of  relative  positions  is  to  increase  or  decrease 
the  number  of  inversions  by  1  according  as  a  is  less  than,  or 
greater  than,  b.  That  is,  the  effect  of  the  interchange  of  two 
adjacent  integers  is  to  increase  or  decrease  the  number  of  in- 
versions by  1. 

Suppose  now  that  there  are  Tc  integers  between  a  and  b. 
Then  b  can  be  brought  to  the  original  position  of  a  by  A;  + 1 
interchanges  of  adjacent  integers,  and  after  this  has  been  done 
a  can  be  brought  to  the  original  position  of  6  by  A:  interchanges 
of  adjacent  integers.  These  interchanges  do  not  affect  the 
relative  positions  of  the  integers  other  than  a  and  b.  The 
desired  interchange  of  a  and  b  has  then  been  brought  about  by 
2  A; -f-  1  interchanges  of  adjacent  integers.  In  general,  some 
of  these  interchanges,  say  x  of  them,  have  each  caused  an 
increase  of  1  in  the  number  of  inversions  and  the  remaining 
2k+l  —  x  have  caused  a  decrease  of  1  each.  The  net  result 
has  been  an  increase  or  a  decrease  of  the  number  of  inver- 
sions equal  to  the  difference  of  these  two  numbers.  But  this 
difference,  which  is  either  2k-{-l  —  2x  or  2x  —  27c—l,  is  an 
odd  number  for  all  possible  values  of  x.  The  theorem  is  there- 
fore proved. 

169o  Inversions  of  letters.  —  In  an  arrangement  of  letters 
of  the  alphabet  the  presence  of  a  letter  before  one  that  pre- 
cedes it  in  the  alphabetical  order  is  also  called  an  inversion. 

Consider  now  the  symbols  aj,  Oa?  ^^3,  •  •  • ;  &i,  &2>  ^3)  •  ••  I  ^i,  Cj,  Cg,  •  •  • ; 
and  so  onj   and  select  a  set  of  these  in  which  neither  any 


DETERMINANTS  203 

letter  nor  any  subscript  occurs  twice.      We  have   then   the 
following 

Theorem.  —  The  number  of  inversions  in  the  letters  when  there 
are  no  inversions  in  the  subscripts  and  the  number  of  inversions 
in  the  subscripts  when  there  are  no  inversions  in  the  letters  are 
both  even  or  both  odd. 

Consider  first  the  special  case  aibzCidc^e^.  Here  there  are  no 
inversions  in  the  letters.  If  we  interchange  dg  and  e^  in  order 
to  bring  the  letter  with  the  greatest  subscript  to  the  last  place, 
we  get  the  arrangement  a^^^^ie^d^.  In  doing  this  we  have 
changed  the  number  of  inversions  in  the  letters  and  the  number 
of  inversions  in  the  subscripts  each  by  an  odd  number.  If  now 
we  interchange  a^  and  eg  in  order  to  bring  the  letter  with  the  next 
to  the  greatest  subscript  to  the  next  to  the  last  place,  we  shall 
again  change  the  number  of  inversions  in  the  letters  and  the 
number  of  inversions  in  the  subscripts  each  by  an  odd  number. 
Finally,  by  the  interchange  of  e^  and  q  we  get  the  arrange- 
ment Cib^e^fi^dr^^  and  this  last  step  changed  the  number  of  inver- 
sions in  the  letters  and  the  number  of  inversions  in  the  sub- 
scripts each  by  an  odd  number.  In  the  final  arrangement  there 
are  no  inversions  in  the  subscripts.  The  number  of  inversions 
in  the  subscripts  has  been  changed  three  times,  by  an  odd 
number  each  time,  and  the  number  of  inversions  in  the  letters 
has  been  changed  the  same  number  of  times,  by  an  odd  number 
each  time.  Hence  the  number  of  inversions  in  the  subscripts 
in  the  original  arrangement  and  the  number  of  inversions  in 
the  letters  in  the  final  arrangement  are  both  odd. 

In  general  we  can  change  the  arrangement  in  which  there 
are  no  inversions  in  the  letters  to  the  arrangement  in  which 
there  are  no  inversions  in  the  subscripts  by  a  certain  number 
of  interchanges  of  letters.  Each  of  these  interchanges  increases 
or  diminishes  the  number  of  inversions  in  the  subscripts  by  an 
odd  number  (§  168),  and  also  increases  or  diminishes  the  num- 
ber of  inversions  in  the  letters  by  an  odd  number.  Hence  the 
number  of  inversions  in  the  letters  when  there  are  no  inver- 


204  COLLEGE  ALGEBRA 

sions  in  the  subscripts  and  the  number  of  inversions  in  the 
subscripts  when  there  are  no  inversions  in  the  letters  are  both 
even  or  both  odd  according  as  the  number  of  these  successive 
interchanges  is  even  or  odd. 

170.  Definition  of  a  determinant.  —  In  an  array  of  n^  num- 
bers in  n  rows  and  n  columns  form  all  possible  products  ofn  fac- 
tors each,  taking  as  factors  one  number,  and  but  one,  from  each 
row  and  column. 

Arrange  the  factors  of  each  product  in  the  order  of  the  columns 
in  which  they  occur  and  change  the  signs  of  those  products  in 
which  the  arrangement  of  the  integers  representing  the  respective 
rows  presents  an  odd  number  of  inversions. 

The  algebraic  sum  of  these  products  with  their  signs  thus  modi- 
fied is  called  a  determinant  of  the  nth  order.  It  is  represented 
by  the  array  inclosed  between  two  vertical  lines. 

Thus,        hi    ^1 1  =  ai62  -  «2&i, 
I  a-z    &2  I 


=  ai&2C3  +  dibzCx  +  azbiCz  —  azb^Ci—  a^biCz  ■—  ciibzC-2, 


ai 

bi    ci| 

a^    hz    C2 

as    bs    cs 

2    4      - 

-1 

3    6 

2 

4    7 

6 

=  2.6.5+3-7-(-l)+4.4.2-4.6.(-l)-2.7.2-3.4.5  =  7. 

Here,  for  example,  the  sign  of  the  product  3  •  4  •  5  is  changed  because 
when  its  factors  are  arranged  in  the  order  of  the  columns  in  which  they 
occur  there  is  an  inversion  in  the  order  of  the  rows  in  which  they  occur, 
this  order  being  2,  1,  3. 

171.  Definitions.  —  The  products  described  in  the  definition 
with  their  proper  signs  are  called  the  terms  of  the  determinant. 

When  a  determinant  is  written  out  in  full  as  the  algebraic 
sum  of  its  terms,  it  is  said  to  be  expanded. 

The  numbers  in  the  array  from  which  the  terms  are  formed 
are  called  the  elements  of  the  determinant. 


DETERMINANTS  205 

172.  Properties  of  determinants.  —  I.  There  are  n\  terms  in 
the  expansion  of  a  determinant  of  order  n. 

This  is  an  immediate  consequence  of  the  fact  that  there  are 
n !  permutations  of  the  n  rows  taken  n  at  a  time  (see  §  104). 

When  n  is  greater  than  3,  n !  is  so  large  that  it  is  not  prac- 
ticable to  find  the  value  of  the  determinant  by  the  direct  pro- 
cess of  expansion.  And  besides  the  number  of  the  terms  there 
is  the  difficulty  of  determining  which  of  the  products  described 
in  the  definition  should  have  their  signs  changed.  These  dif- 
ficulties can  be  avoided  by  the  application  of  some  of  the  fol- 
lowing properties  of  determinants. 

II.  A  determinant  is  not  changed  if  the  corresponding  columns 
and  rows  are  interchanged. 

Since  each  term  of  a  determinant  of  order  n  is  the  product, 
with  a  possible  change  of  sign,  of  n  factors,  no  two  of  which 
occur  in  the  same  row  or  the  same  column,  it  is  evident  that 
the  interchange  of  corresponding  rows  and  columns  will  have 
no  effect  on  the  terms  except  perhaps  to  change  the  signs  of 
some  of  them. 

If  the  original  form  of  the  determinant  is 

«!      6i      Ci    •  • 
0,2      &2      C2    •  • 


dn       bn      Cn   •• 

any  of  its  terms  is  a  product  of  n  factors  so  arranged  that 
there  are  no  inversions  in  the  letters  and  the  sign  of  this  prod- 
uct is  changed  or  not  according  as  the  number  of  inversions 
in  the  subscripts  is  odd  or  even.  The  corresponding  term  of 
the  new  determinant  is  the  product  of  the  same  factors  so  ar- 
ranged that  there  are  no  inversions  in  the  subscripts  and  the 
sign  of  the  product  is  changed  or  not  according  as  the  number 
of  inversions  in  the  letters  is  odd  or  even.  But  by  the  theo- 
rem of  §  169,  the  number  of  inversions  in  the  subscripts  is 


206 


COLLEGE  ALGEBRA 


ai 

&i 

Cl 

052 

62 

C2 

= 

as 

63 

C3 

here  odd  or  even  according  as  the  number  of  inversions  in  the 
letters  is  odd  or  even.     Hence  the  determinant  is  not  changed 
by  the  interchange  of  corresponding  rows  and  columns. 
The  student  should  verify  directly  that 

<Xl  0,2  Q!3 
h\  &2  ^3 
Cl       C,2       C3 

It  follows  from  II  that  for  every  theorem  concerning  the 
rows  of  a  determinant  there  is  an  analogous  theorem  concern- 
ing its  columns. 

III.  If  all  the  elements  of  a  row  (or  column)  of  a  determinant 
are  multiplied  by  the  same  number,  the  value  of  the  determinant  is 
multiplied  by  this  number. 

This  follows  from  the  fact  that  one,  and  only  one,  element 
from  this  row  (or  column)  is  a  factor  of  every  term  of  the  ex- 
pansion of  the  determinant. 

For  example, 


=kaib-2C3+lca2b3Ci  -\-  Tcazb\C2  —  kazb^ci  —  ka\bzC2  —  ka^biCz 


«1 

61 

Cl 

ka2 

kb2 

kC2 

«3 

bz 

Cz 

=  k  {aibzCa  +  a^bsCi  +  «3&iC2  —  a3&2Ci  -  ai&3C2  — «26iC3)  =  k 


bi 

Cl 

&2 

C2 

h 

C3 

IV.    The  sign  of  a  determinant  is  changed  if  tico  of  its  rows 
(or  columns)  are  interchanged. 

This  is  an  immediate  consequence  of  the  definition  of  a 
determinant  and  the  theorem  of  §  168. 


For  example 

1  bi    C] 
J     63    cs 

2  62    Cj 


ai&3C2  +  a3&2Ci  +  a2&iC8  —  a2&3Ci  —  asftiCj  —  016203 


=  —  (ai62C3  +  dibsCi  4-  (iabiC2  —  a3b2Ci  —  aibsC2  —  (i2biCs)  =  — 


ai 

bi 

Cl 

a2 

62 

C2 

as 

?>3 

C3 

DETERMINANTS 


207 


V.  The  value  of  a  determinant  is  zero  if  the  corresponding  ele- 
ments of  two  of  its  rows  (or  columns)  are  the  same. 

Let  D  represent  the  determinant. 

Then  by  IV  D  is  changed  into  —  D  by  the  interchange  of 
any  two  rows  (or  columns).  On  the  other  hand,  if  these  two 
rows  (or  columns)  are  the  two  identical  ones,  D  is  not  affected 
by  their  interchange.     Hence 

D  =  -D, 
2D  =  0, 
and  D  =  0. 

For  example, 

«i    t>i    ci  I 

ai    bi    ci  I  =  aibics  +  ai&sCi  -f  a^hiCi  —  asbiCi  —  aibsCi  —  aibiCs  =  0. 

as    63    C3 1 

VI.  Hie  value  of  the  determinant  is  zero  if  the  elements  of  any 
row  {or  column)  are  m  times  the  corresponding  elements  of  any 
other  roiv  (or  column). 

We  know  from  III  that  the  value  of  such  a  determinant  is 
m  times  the  value  of  a  determinant  the  corresponding  elements 
of  two  of  whose  rows  (or  columns)  are  the  same,  and  we  know 
from  V  that  the  value  of  this  latter  determinant  is  zero. 


For  example, 


ai  &i  Ci 
a2  &2  C2 
ma2    W162    niC2 


—  maibiCi  +  ma2&2Ci  +  maibiCi 
—  maib^Ci  —  maibiCi  —  ma2&iC2  =  0. 


VII.  If  two  determinants  are  the  same  except  possibly  for  the 
elements  of  a  certain  row  (or  column)^  their  sum  is  equal  to  a  de- 
terminant in  which  the  elements  of  this  row  (or  column)  are  the 
sum.s  of  the  corresponding  elements  of  the  two  determinants  and  the 
elements  of  the  other  roivs  are  the  same  as  in  the  two  determinants. 


For  exaa 

iple, 

ai    61    ci 

a'l    &i    ci 

«i  +  a'l 

bi    ci 

0,2       &2       C2 

+ 

a'2       &2      C2      = 

a2  +  a'2 

62    C2 

as    bs    Cs 

a's    bs    Cs 

as  +  a's 

bs    Cs 

208 


COLLEGE  ALGEBRA 


That  this  is  so  is  obvious  when  we  consider  that  every  term  of 
the  third  determinant  is  the  sum  of  two  parts  which  are  respec- 
tively the  corresponding  terms  of  the  first  two  determinants. 

VIII.  The  value  of  a  determinant  is  not  changed  if  each  ele- 
ment of  any  row  (or  column)  multiplied  by  any  number  m  be 
added  to  the  corresponding  elements  of  any  other  row  (or  column). 

By  VII  the  resulting  determinant  is  equal  to  the  sum  of  two 
determinants  of  which  one  is  the  original  determinant  and  the 
other  has  the  value  zero  by  VI. 

For  example, 
ai  +  mci    61    ci 

Gz  +  WIC2      62      C2 

as  +  mcs    63    C3 

173.  Minors  of  a  determinant. — If  in  a  determinant  of 
order  n  we  omit  the  elements  of  any  row  and  any  column,  the 
remaining  elements  with  their  relative  positions  unchanged 
form  a  determinant  of  order  n  —  1  which  is  called  the  minor  of 
the  element  in  this  row  and  this  column. 

Thus,  if  in  the  determinant 


ai    61    Ci 

= 

tti      &2      Co 
0,3       &3       C3 

+ 

mci    61    Ci 

ai    61    ci 

TOC2    62    C2 

= 

a2    &2    C2 

WC3       &3       C3 

«3    h    C3 

«i 

&i 

Cl 

di 

a2 

62 

C2 

d2 

as 

63 

C3 

ds 

a4 

64 

C4 

d4 

we  omit  the  elements  of  the  second  row  and  the  third  column  the  remain- 
ing elements  form  the  determinant 

Oi    61    di 

as    63    ds 

a^    &4    d^ 
which  is  the  minor  of  C2. 

If  an  element  of  a  determinant  is  represented  by  a  small 
letter  with  a  subscript,  we  represent  its  minor  by  the  corre- 
sponding capital  letter  with  the  same  subscript. 


DETERMINANTS 


209 


ai    &i 

Cl 

a2  &2 

C2 

as  bs 

C3 

62   C2 

—  a^ 

61  Cl 

+  «3 

&1   Cl 

63   C3 

&3   C3 

&2   C2 

174.   The  expansion  of  a  determinant  of  the  third  order  can 
be  arranged  as  follows : 


=  ai62C3  +  ciibiCi  +  a3&iC2  —  03&2C1  —  O2&1C3  —  ai&3C2 
=  ai(&2C8  —  &3C2)  —  ttiibiCs  —  63C1)  +  a3(&iC2  —  62C1) 
=  ai 
=  ai-4i  —  a2^2  +  «3^3- 


That  is,  a  determinant  of  the  third  order  can  be  expressed  in 
terms  of  determinants  of  the  second  order.  In  a  similar  way 
a  determinant  of  order  n  can  be  expressed  in  terms  of  deter- 
minants of  order  n  —  1.  The  following  theorem  states  in  de- 
tail how  this  can  be  done. 

Theorem.  —  Multiply  the  elements  of  any  row  (or  column)  of  a 
determinant  by  their  respective  minors,  and  if  the  sum  of  the 
number  of  the  roiv  and  the  number  of  the  column  in  which  the  ele- 
ment occurs  is  odd,  change  the  sign  of  the  resulting  product.  The 
algebraic  sum  of  these  products  with  their  signs  thus  changed  is 
equal  to  the  determinant. 

Represent  the  determinant  by  D,  the  element  in  the  upper 
left  hand  corner  by  Oj,  and  any  other  element  by  s^. 

1.  If  the  elements  in  the  terms  of  D  and  Ai  are  arranged  as 
described  in  §  170,  the  coefficient  of  a^  in  any  term  of  Z>  is  a 
term  of  Ai,  since  the  row  and  column  in  which  ai  occurs  pre- 
cede all  the  other  rows  and  columns.  This  conclusion  is  based 
upon  the  fact  that  in  any  arrangement  of  a  series  of  integers 
beginning  with  1  the  number  of  inversions  is  the  same  as  it  is 
in  this  arrangement  with  the  1  omitted.  Thus  135462  and 
35462  contain  the  same  number  of  inversions. 

Conversely,  any  term  of  Ai  is  the  coefficient  of  aj  in  some 
term  of  D. 


210 


COLLEGE  ALGEBRA 


62C3  —  53C2,  which  is 


ai 

&i 

Cl 

det 

erm 

inant 

az 

&2 

C2 

«3 

63 

C3 

&2 

C2 

,  or  Ai. 

&3 

C3 

the  coefficient  of  ai  is 


2.  Consider  the  element  s^  in  the  ith  row  and  Jth  column  of 
D.  It  can  be  brought  to  the  first  row  by  i  —  1  interchanges  of 
adjacent  rows,  and  then  to  the  first  column  by  J  —  1  inter- 
changes of  adjacent  columns,  Avithout  affecting  the  relative 
positions  of  the  elements  of  D  not  contained  in  the  original  ith 
row  or  the  original  Jth  column.  The  effect  of  these  inter- 
changes is  to  change  the  sign  of  the  determinant  (^  —  1  4-J—  1) 
times;  that  is,  (i-\-j  —  2)  times.  This  number  is  odd  or  even 
according  as  i-\-j  is  odd  or  even.  If  then  D'  denotes  the 
determinant  resulting  from  these  changes, 

iy=(-iy+JD. 

If  we  multiply  both  members  of  this  equation  by  (— !)'+■', 

we  get  ( _  iy+^-  2/  =  (_  iyi+2j  d  =  D, 

since  (-If +2^=1. 

Since  now  the  relative  positions  of  the  elements  not  in  the 
ith  row  or  the  Jth  column  of  D  are  not  affected  by  these 
changes,  the  minor  of  s^  in  U  is  the  same  as  its  minor  in  D. 
And  from  (1)  we  see  that  the  coefficient  of  s^  in  jy  is  Si- 
Hence  the  coefficient  of  s^  in  Z)  is  (—  Vf'^^  Si. 


For  example,  in  the  determinant 


63.     Here  i  =  3  and  j  =  2. 


Now 


«i  hi 
a^  62 
as    ^3 


0^2 

as 
a4 

C3 

Cl 

C2 
C4 


consider  the  element 


63 

as 

Cz 

dz 

61 

ai 

Cl 

dx 

62 

a2 

C2 

d2 

64 

a4 

C4 

d. 

DETERMINANTS 


211 


From  (1)  we  know  that  the  coefficient  of  63  in  this  last  determinant  is 
and  therefore  the  coeflBcient  of  63  in  the  original  deter- 


ai    ci    di 

ai    C2    d-i 

,  and  theref( 

ai    C4    di 

ai    ci    di 

minant  is  — 

a2    C2    di 

as    C3    di 

We  shall  get  all  the  terms  in  the  expansion  of  D  by  taking 
the  sum  of  all  the  terms  containing  each  of  the  elements  of  a 
given  row  or  column,  and  no  term  will  appear  more  than  once, 
since  no  term  in  the  expansion  of  D  can  contain  two  elements 
from  the  same  row  or  column. 

This  proves  the  theorem. 

In  order  to  see  clearly  the  meaning  of  this  theorem  consider  its  appli- 
cation to  a  determinant  of  the  fourth  order : 


ai 

&i 

Cl 

di 

a2 

62    C2 

do 

as 

&3       C3 

di 

ai 

hi    C4 

di 

61 

Cl 

=  03 

&2 

C2 

bi 

Ci 

+  C3 


ds 


175.  By  means  of  successive  applications  of  the  theorem  of 
the  preceding  article  we  can  make  the  expansion  of  a  determi- 
nant depend  finally  upon  the  expansion  of  a  series  of  determi- 
nants of  the  third  order.  This  process,  simplified  by  means 
of  the  properties  given  in  §  172  in  a  way  we  shall  explain  by 
means  of  examples,  is  the  one  most  frequently  used  for  the 
expansion  of  a  determinant. 

Example  1.  —  Expand  the  determinant 
2    12         1 
3     1         4 
7     7     3         2 
G    G     5     -1 


212 


COLLEGE  ALGEBRA 


If  we  subtract  the  elements  of  the  second  column  from  the  correspond- 
ing elements  of  the  first  column,  the  resulting  determinant  is,  by  §  172, 
VIII,  where  w  =  -  1,  equal  to  D.     Hence 


1     1     2 

1 

D  = 

0    3     1 
0     7     3 

4 

2 

• 

,  by  §  174, 

0    6    5 

-1 

3     1 

4 

0     1 

0 

D  = 

7     3 

2 

= 

-2    3 

-10 

=  _  42  +  90  =  48 

6     5 

-1 

-9     5 

-21 

The  last  determinant  here  is  obtained  from  the  preceding  one  by  an 
application  of  §  172,  VIII. 

The  advantage  of  the  preliminary  transformation  is  due  to  the  fact 
that  it  puts  D  into  a  form  in  which  all  but  one  of  the  elements  of  a  cer- 
tain column  are  zero.  In  the  expansion  of  a  determinant  in  such  a  form 
by  §  174  all  the  terms  except  the  first  one  are  equal  to  zero. 


Example  2 

.— 

Expand  the  determinant 

9     13     17     -4 

D  = 

18    27     35     -8 
30    44     21        10 
12     15      0         2 

• 

9 

13     17     -  4 

9     13           4-4 

D  = 

0 

30 
12 

1       1         0 
44    21        10 
15      0         2 

= 

0       1           0         0 
30     44     -23       10 
12     15-15         2 

9 

4     -4 

3           4-4 

3 

4 

0 

= 

30 

-23       10 

=  3 

10     -  23        10 

=  3 

10 

-23 

-13 

12 

-15         2 

4-15         2 

4 

-15 

-13 

=  3(897  -  208  -  585  +  520)  =  1872. 


Here  we  multiplied  the  elements  of  the  first  row  of  the  original  deter- 
minant by  2  and  subtracted  the  products  from  the  corresponding  elements 


DETERMINANTS 


213 


of  the  second  row.  Then  we  subtracted  the  elements  of  the  second  column 
of  the  resulting  determinant  from  the  corresponding  elements  of  the  third 
column. 

If  the  determinant  has  an  element  equal  to  ±1,  we  can  easily,  by 
applying  §  172,  VIII,  make  all  the  other  elements  in  the  same  row,  or 
the  same  column,  zero.  Then  by  the  theorem  of  §  174  the  determinant 
is  equal  to  a  determinant  of  lower  order. 

If  the  determinant  has  no  element  equal  to  ±  1,  a  suitable  application 
of  §  172,  VIII,  will  transform  it  into  one  with  at  least  one  element  equal 
to  ±  1. 

When  we  get  to  determinants  of  the  third  order,  it  is  best  to  expand 
them  in  full  unless  there  is  an  obvious  way  to  shorten  the  process. 

The  student  should  always  be  on  the  lookout  to  see  if  the  elements  of 
any  row  (or  column)  are  equal  to  w  times  the  corresponding  elements  of 
any  other  row  (or  column).  In  this  case  the  determinant  equals  zero 
(§  172,  VI). 

If  the  elements  of  any  row  (or  column)  are  integers  with  a  common 
factor,  §  172,  III,  can  be  applied  to  advantage. 

176.  Theorem.  —  If  the  elements  of  a  determinant  D  are 
polynomials  in  x,  and  if  D  =  0  when  x  =  a,  then  x—  a  is  a  factor 
of  the  expansion  of  D. 

This  is  an  immediate  consequence  of  the  Factor  Theorem 
(§  142,  Cor.),  since  the  expansion  of  D  is  a  polynomial  in  x. 


For  example,  if 


D  = 


x^  X  1 
4  2  1 
1       1     1 


it  is  evident  that  Z)  =  0  when  a;  =  1  and  when  a;  =  2  (§  172,  V).  Hence 
D  is  divisible  by  x  —  1  and  by  a;  —  2.  Moreover  it  is  clear  from  §  174 
that  D  is  of  the  second  degree  in  x,  and  that  the  coefficient  of  x^  is 


2     1 
1     1 


Hence 


X2 

X 

1 

4 

2 

1 

1 

1 

1 

=  (^-l)(^-2). 


214 


COLLEGE  ALGEBRA 


EXERCISES 

1.  How  many  inversions  are  there  in  the  arrangement 
1,  7,  5,  8,  6,  2,  4,  3  ?     In  the  arrangement  7,  5,  8,  6,  2,  4,  3  ? 

2.  How  many  inversions  are  there  in  the  subscripts  of 
ag  ^5  Cq  di  67  /4  ^3?  How  many  inversions  in  the  letters  of 
^1  «2  93  fi  ^5  Co  67  ?     Compare  §  169. 

3.  Determine  by  inspection  whether  the  following  two 
determinants  are  equal  or  not: 


3 

-4 

0 

1 

3 

6 

1 

-6 

6 

2 

7 

9 

-4 

2 

3 

0 

1 

3 

8 

4 

> 

0 

7 

8 

7 

6 

0 

7 

3 

1 

9 

4 

3 

4.    Show  without  expanding  the  determinants  that 


3           0  14  2 

7           5  4  9 

8-10  1  6 

5           8  3  12 


-f- 


3           0  2  14 

7           5  9  4 

8-10  6  1 

5           8  12  3 


0. 


5.    Express  the  following  sum  as  a  single  determinant  of 
the  fourth  order : 


15 

2 

6 

1 

5 

0 

4 

8 

+ 

2 

rr 
i 

3 

10 

8 

8 

2 

2 

0 

2 

6 

1 

5 

0 

4 

8 

7 

7 

3 

10 

2 

8 

2 

2 

Find  the  values  of  the  following  determinants : 


6. 


3  15  16  29 
6  7  33  14 
6  30  32  58 

4  2  6  6 


7. 


3  4-2 
4-3  8 
2         8-3 


DETERMINANTS 


215 


3 

4 

10 

0 

-6 

8 

0 

0 

5 

0 

0 

0 

10. 


1 

0 

0 

0 

3 

4 

5 

2 

3 

2 

4 

1 

1 

1 

2 

1 

4  7    6  5 

2  13  7 

2  5     3  4 

14  4     6  5 


11. 


0     f 


Factor  the  following  determinants : 


12. 


a^  a^  X  1 
1111 
8     4     2     1 

8  4  2.       1 

2T       ST         3       ^ 


13. 


3  +  a;     4:  —  x     x-\-l 
5  2  3 

0  6  1 


14.    Show  that 


111 

X      y       z 

X^       y^       2;2 


is  divisible  hy  x  —  y,y  —  z,  and  z  —  x. 


15.    Show  that 


X 

2/ 

1 

2 

3 

1 

1 

4 

1 

=  0 


when  x=2,  2/  =  3 ;  and  also  when  x  =  —  l,  2/  =  4. 


16.    Show  that 


y   1 

2    1 
5     1 


=  0 


is  the  equation  of  the  straight  line  through  the  points  (—1,  —  2) 
and  (3,  5). 


216 


COLLEGE  ALGEBRA 


17.   Find  three  sets  of  values  of  x,  y,  and  z  for  which 

X  y       z    \ 

5  3-21 

4  111 

0  0       6    1 


=  0. 


177.  Solutions  of  systems  of  linear  equations  by  means  of 
determinants. —  It  was  stated  at  the  beginning  of  this  chapter 
that  determinants  of  order  n  could  be  used  in  the  solution  of 
systems  of  n  linear  equations  in  n  unknowns.  We  are  now  in 
a  position  to  see  in  detail  how  this  is  done. 

We  shall  consider  the  solution  of  a  system  of  four  linear 
equations  in  four  unknowns.  The  method  used  is  applicable 
to  the  general  case. 

Let  the  given  equations  be 

axx  +  hxy  +  c^z  +  d\V3  =  A^i,  (1) 

a^x  +  622/  +  Ciz  -}-  d%w  =  ki,  (2) 

aax  -\-  bsy  +  Csz  +  dsw  =  ks,  (3) 

ttiX  +  biy  +  c^z  +  diW  =  k^.  (4) 

The  determinant  formed  by  the  coefficients  of  the  unknowns, 
that  is, 


«1 

bi 

Cl 

di 

aa 

&2 

C2 

d2 

as 

63 

cs 

dz 

a4 

h 

C4 

d. 

is  called  the  determinant  of  the  system  of  equations.  We 
shall  represent  it  by  D,  and  shall  assume  that  the  equations 
are  such  that  D=^0.  If  there  is  a  set  of  values  of  x,  y,  z,  and 
w  that  satisfy  these  equations,  these  values  can  be  found  in 
the  following  way : 

Multiply  each  member  of  equations  (1),  (2),  (3),  and  (4)  by 
Aij  —A2,  As,  and  —A^  respectively.     This  gives 

QiAix  +  biAiy  +  c\Aiz  +  diA\w  = 

—  QiAiX  —  b^A^y  —  CiA<iz  —  diA^w  =  ■ 
azAzX  -f  bzAzy  +  c^Azz  +  d&A-iW  = 

—  a^A^x  —  64^44^  —  C4A4Z  —  diAiW  = 


kiAi, 

(5) 

kiAi, 

(6) 

kzAz, 

(7) 

kUi. 

(8) 

DETERMINANTS 


217 


Putting  the  sum  of  the  left  members  of  equations  (5),  (6), 
(7),  and  (8)  equal  to  the  sum  of  their  right  members,  we  get 

x{aiAi  —  a2^2  +  azAs  —  a^A^)  +  yibiAi  —  h^Ai  +  hzAz  —  64-44) 
+  z{cxAx  —  C2A2  +  CsAs  -  04^4)  • 
+  w(diAi  —  d-zAi  +  dsAs  —  d^A^) 
—  kiAi  —  kiA^  +  kzAz  —  k^A^.  (9) 

Now  the  coefficient  of  x  in  equation  (9)  is,  by  §  174,  equal  to 
Df  and  the  coefficients  of  y,  2;,  and  w  are 


di  61  Ci  di 

d'2  62  C2  di 

ds  63  C3  ds 

d^  64  C4  (?4 


respectively.     But  these  last  three  determinants  are,  by  §  172, 
V,  equal  to  zero. 

Hence  this  equation  can  be  written  in  the  form 


61 

61 

ci  di 

62 

62 

C2      d2 

68 

63 

Cs    ds 

1 

64 

64 

C4  di 

Cl   61   Ci 

(^1 

C2  62  C2 

d2 

Cs      63  Cs 

ds 

1 

C4  64  C4 

di 

Da;  =  kiAi  —  k^A^  +  ksAs  —  k^A^. 

In  a  similar  way  we  get 

Dy  =-  kiBi  +  k2B2  -  ksBs  +  kiB^, 
Dz  =  ^•l(7l  —  A:2C2  +  ksCs  —  k^Ci, 
Dw  =  -  kiDi  -{-k2D2  -  ksDs  +  A;4i>4. 


(10) 


(11) 
(12) 
(13) 


Now  the  right  members  of  these  four  equations  are  equal  to 


kx    61  Cl  di 

k2     62  C2  d2 

ks    63  Cs    dz 

> 

ki    64  Ci    di 

«i 

A-i 

Cl 

dx 

rt2 

A:2 

C2 

d2 

as 

A:3 

Cs 

ds 

1 

04 

ki 

Ci 

di 

ax 

bx    kx    dx 

a2 

62  k2    d2 

as 

63  ks    ds 

1 

Qi 

hi    ki    di 

ax 

bx 

cx 

kx 

a2 

62 

C2 

ki 

as 

63 

Cs 

ks 

ai 

64 

Ci 

ki 

respectively. 

Therefore,  if  there  is  a  set  of  values  of  x,  y,  z,  and  w  that 
satisfy  equations  (1),  (2),  (3),  and  (4),  these  values  must  be 
given  by  the  formulae 


218 


COLLEGE  ALGEBRA 


^•l 

hi 

Cl 

^1 

k2 

62 

C2 

d2 

ks 

&3 

C3 

dz 

k. 

&4 

C4 

d. 

ai 

ki 

Cl 

c?i 

a2 

k2 

C2 

d2 

as 

ks 

C3 

<?3 

ai 

k. 

C4 

d4 

D 


ai 

61 

^1 

di 

a2 

62 

k2 

d2 

as 

&3 

ks 

ds 

a4 

&4 

k. 

d4 

D 


«1 

&1 

Cl 

ki 

«2 

62 

C2 

k2 

«3 

63 

C3 

ks 

a4 

&4 

C4 

k. 

D 


(14) 


It  is  now  necessary  to  show  that  these  values  do  satisfy 
equations  (1),  (2),  (3),  and  (4).  If  we  substitute  these  values 
for  X,  y,  Zy  and  w  respectively  in  the  left  member  of  (1),  we  get 


Ol 


kx  h  Cl  di 

«i  ki  Cl  di 

ai  61  ki  di 

ai  bi  Cl  ki 

ki    62    C2    d2 

ks  63  C3  ds 

+  6. 

a2  ^•2  C2  d2 
«3   ^'3   C3   ds 

+  Ci 

a2  &2  k2  d2 
as  63  ^3  ds 

+  di 

a2    &2   C2   k2 

as  63  C3  ks 

kA   &4   C4   ^4 

ai  k4  ca  d4 

Ga  bA  kA  dA 

ttA  bA  Ca  kA 

And  this,  when  arranged  with  reference  to  k^,  k^,  k^,  and  kA 
becomes 

A:i(ai^i  -  61-Bi  +  CiCi  -  diD{)  +  k2{-  aiAz  +  61-B2  -  C1C2  +  di2>2) 

+  ksiaiAs  -  biBs  +  c^Cs  -  diDs) 

+  kA(—  OiAa  +  biBA  —  CiCa-\-  <?il>4) 
_ 


In  the  numerator  of  this  fraction  the  coefficient  of  ki  is  D 
and  the  coefficients  of  A^s,  k^,  and  kA  -are 


«1 

61 

Cl 

di 

ai 

&i 

Cl 

d. 

as 

&3 

C3 

ds 

ttA 

?>4 

C4 

dA 

ai 

61 

Cl 

d. 

a2 

&2 

C2 

d2 

a\ 

&i 

Cl 

di 

' 

aA 

64 

C4 

dA 

ai 

bi 

Cl 

d. 

a2 

62 

C2 

d2 

as 

63 

C3 

ks 

ai 

&i 

Cl 

di 

respectively.  Each  of  these  last  three  determinants  is  equal 
to  zero.  Hence  the  fraction  is  equal  to  ki.  That  is,  these 
values  of  x,  y,  z,  and  w  satisfy  equation  (1).     In  a  similar  way 


DETERMINANTS  219 

it  can  be  shown  that  these  values  also  satisfy  equations  (2), 
(3),  and  (4). 

The  student  should  observe  that  these  formulae  for  the  solu- 
tion of  equations  (1),  (2),  (3),  and  (4)  are  closely  analogous  to 
the  formulae  for  the  solution  of  systems  of  two  or  three  linear 
equations  in  two  or  three  unknowns  respectively.  They  apply 
with  slight  and  obvious  modifications  to  the  solutions  of  sys- 
tems of  n  linear  equations  in  n  unknowns. 

178.  Inconsistent  equations.  —  The  argument  by  means  of 
which  these  solutions  were  obtained  breaks  down  in  case  i)=0. 
In  this  case  the  left  members  of  equations  (10),  (11),  (12),  and 
(13),  §  177,  are  all  zero.  Hence,  unless  the  numerators  of 
formula  (14)  are  all  zero,  we  have  a  contradiction,  and  the 
original  equations,  (1),  (2),  (3),  and  (4),  are  inconsistent.  This 
argument  can  be  extended  to  a  system  of  n  linear  equations  in 
n  unknowns.  We  conclude  therefore  that  if  the  determinant 
D  of  a  system  of  n  linear  equations  in  n  unknowns  is  zero,  the 
equations  are  inconsistent  unless  all  the  determinants  in  the 
formulce  analogous  to  (14)  are  zero. 

Example.  —  Show  that  the  equations 

-2x  +  4y  +  5z-{-2w  =  3, 

Sx-6y  —  6z  +  iw  =  6, 

8x  +  6y-\-z  +  5w  =  4:j 
are  inconsistent. 

The  equations  may  be  inconsistent  when  D  and  all  the  de- 
terminants in  the  formulae  analogous  to  (14)  are  zero.  This 
question  is,  however,  too  complicated  to  be  discussed  here. 

179.  Homogeneous  equations.  —Equations  (1),  (2),  (3),  and 
(4)  are  said  to  be  homogeneous  when  their  right  members  are 
all  zero.  In  this  case,  ii  D^O,  the  only  values  of  the  unknowns 
that  satisfy  the  equations  are  ic  =  0,  2/=0,  2  =  0,  and  to  =  0. 
If  Z)  =  0,  there  are  other  solutions.  An  analogous  thing  holds 
for  a  system  of  n  linear  homogeneous  equations  in  n  unknowns. 


220  COLLEGE  ALGEBRA 

EXERCISES 

Solve   the    following   systems   of    equations   by   means    of 
formulae  (14)  or  similar  formulae : 

7.  Sx-{-y-{-w=:20, 
8z-^6x-\-w  =  ^0, 
Sz-\-x-\-4:y  =  30, 
5z  +  8y  +  Sw  =  50, 

8.  .4  +  2J5  +  3(7+42)  =  l, 
4^+54-2  (7+3  J9+3  =  0, 
3  A+A  B-\-C-j-2  D+6  =  0, 
2^+35+4  C+D-f  2  =  0. 

9.  A+2  B-hS  C-D-2  E=5, 
2^+J5+3  a-2Z>+^=4, 
B-h2C-2D-{-E=2, 
-B-C-\-2D  +  E  =  0, 

2B  +  SC-\-D-E  =  e. 

4.  2x-{-Sy-z-^S  =  0j 

2x-3y  +  3z  =  2, 
—  x-{-2y-i-5z=:5. 

5.  3^  +  2^-0+D=2, 
A-{-3B+G-D  =  5, 
A-{-B  =  4:,  11.  3x  +  2y-5z=:5, 
0+D  =  6,  4.x-6y-\-2z=:-10, 

2x-^10y-12z  =  S. 

6.  —4:X-\-5y-^4:Z  —  3w  =  0, 

jx-{-y-\-z  +  w==34:,  12.  A-5B-6C=2, 

4:X-10y-\-4.z-w  =  6,  -3^-45  +  100=7, 

x-6y-Sz-{-5w=-ld.  10J.  +  25-36O  =  -l. 


1. 

2x-y-^z  =  5, 

y  —  2z  —  5w  =  —  S, 

x  +  y  +  z-{-iv  =  2, 

x  —  2y-{-5iv  =  6. 

2. 

5x  —  2y  +  z  +  3w  —  7  =  0, 

2a;  +  4?/  +  32  +  2w  =  8, 

x  —  y-~z-{-2v  =  4:f 

3x  +  y  +  5z-\-4:W  =  9. 

3. 

x-\-y  =  a, 

y  +  z^h, 

z-\-w=  c, 

2w-{-x=^d. 

1.  x  +  y-\-z 
x  —  y-\-z 
2x->f-3y- 

—  w  = 

-2W: 

=  a, 
=  5, 
=  Z, 

y-\-z  =  m. 

CHAPTER  XV 

INEQUALITIES 

.  180.  Definition  of  inequality.  —  We  have  considered  in  some 
detail  how  to  identify  numbers  that  are  described  by  means  of 
equations.  In  certain  important  problems,  however,  the  infor- 
mation we  are  given  concerning  the  numbers  involved  is  more 
indefinite  than  that  contained  in  equations.  It  merely  tells  us 
that  one  thing  i^  greater  or  less  than  another.  Such  a  state- 
ment is  called  an  inequality. 

Inequalities  are  used  only  in  connection  with  real  numbers, 
and  the  real  number  a  is  greater  than  the  real  number  h  if 
a— 6  is  positive,  and  a  is  less  than  6  if  a— 6  is  negative.    (§  3.) 

The  expression  a  ^  6  means  that  a  is  greater  than,  or  equal 
to,  6,  and  a^h  means  that  a  is  less  than,  or  equal  to,  h. 

Two  inequalities  like  a>h  and  c > d,  in  each  of  which  the 
greater  number  appears  on  the  same  side,  are  said  to  be  alike  in 
sense.  On  the  other  hand,  a>6  and  c<d  are  said  be  opposite 
in  sense. 

181.  Conditional  and  unconditional  inequalities.  —  Some  in- 
equalities are  satisfied  by  all  real  numbers  and  are  therefore 
called  unconditional  inequalities.  They  are  analogous  to  iden- 
tical equations. 

Thus  x^  +  1  >  0  is  an  unconditional  inequality.  Such  a  description 
gives  us  no  information  whatever  about  the  numbers  involved. 

Many  inequalities  on  the  other  hand  are  satisfied  only  by 
certain  numbers.     These  are  called  conditional  inequalities. 

Thus  a;  +  1  >  0  is  a  conditional  inequality,  since  not  every  real  number 
when  added  to  1  gives  a  sum  greater  than  0. 

221 


222  COLLEGE  ALGEBRA 

182.  Properties  of  inequalities.  —  In  order  to  identify  as 
closely  as  possible  the  numbers  described  by  means  of  an  in- 
equality we  proceed  very  much  as  in  the  solution  of  an  equa- 
tion. The  justification  of  such  a  procedure  is  found  in  the 
following  theorems : 

I.  Tlie  sense  of  an  inequality  is  not  changed  if  both  sides  are 
increased  or  decreased  by  the  same  number. 

For  if  a>b, 

then  a  —  b  =  7c,  where  Zc  is  a  positive  number, 

and  a-]-n  —  b  —  n  =  k,  w^here  n  is  any  real  number,  positive  or 

negative ; 

or  (a  4-  n)  —  (6  -f  n)  =  k. 

Hence  a-\-n>b  -\-n. 

This  demonstration  applies,  of  course,  to  inequalities  of  the 
opposite  sense  since  the  statement  that  a>^  is  equivalent  to 
the  statement  that  b<a. 

It  follows  that  terms  can  be  transposed  from  one  side  of  an 
inequality  to  the  other  by  changing  their  signs,  as  in  the  case 
of  equations. 

Thus,  the  conditional  inequality  a;  +  1  >  0  that  was  referred  to  in 
§  181  is  equivalent  to  the  inequahty  x >—  1. 

II.  The  sense  of  an  inequality  is  not  changed  if  both  sides  are 
multiplied  or  divided  by  the  same  positive  nurnber. 

For  if  a  >  b,  and  m  is  any  positive  number, 

then  a  —  b  =  1c,  where  A;  is  a  positive  number, 

and  am  —  bm  =  km. 

But  since  k  and  m  are  both  positive,  km  is  positive,  and 
therefore  am  >  bm. 

III.  The  sense  of  an  inequality  is  reversed  if  both  sides  are 
multiplied  or  divided  by  the  same  negative  number. 

The  proof  of  this  is  left  to  the  student. 


INEQUALITIES  >  223 

It  follows  from  Theorem  III  that  the  sense  of  an  inequality- 
is  reversed  if  the  signs  of  all  its  terms  are  changed. 

Thus,  if  a  >  ft, 

then  —a<  —  b. 

183.  Conditional  inequalities.  —  For  our  purpose  the  most 
important  conditional  inequalities  are  those  involving  one  un- 
known to  the  first  or  second  degree.  These  are  called  linear 
and  quadratic  inequalities  respectively. 

By  Theorem  I  of  the  preceding  article,  every  inequality  in- 
volving one  unknown  can  be  put  into  the  form  f(x)  >  0. 

If  f(x)  is  linear  in  x,  the  inequality  is  of  the  form 

ax  +  b>0. 

Hence  a;  > , 

a 

or  x< , 

a 

according  as  a  is  positive  or  negative. 

If  f(x)  is  a  quadratic  in  x,  we  have 

ax^  -f-  &a;  -}-  c  >  0. 
Now  by  §  148,    ax^ -{- bx -\- c  =  a  (x  —  r^)  (x  —  rg), 
where  Vi  and  rg  are  the  roots  of  the  equation 
aoi^  -\-bx-\-c  =  0. 

If  Vi  and  rg  are  imaginary,  ax^  -i-bx-^c  has  the  same  sign 
for  all  values  of  x,  since  if  it  had  opposite  signs  for  x  =  l  and 
x  =  m,  the  equation  would  have  a  real  root  between  I  and  m 
(§  146).  This  sign  is  the  same  as  the  sign  of  c  inasmuch  as 
ax^ -{-bx-{-c  =  c  when  x  =  0. 

Since  the  condition  for  imaginary  roots  is  (§  71)  6^—  4  ac<  0, 
we  see  that  a  and  c  must  have  the  same  sign  when  the  roots 
are  imaginary.  Hence  in  this  case  ax^  -\-bx-{-c  has  the  same 
sign  as  a  for  all  values  of  x. 


^ 


224 


COLLEGE  ALGEBRA 


If  Vi  =  r2,  (6^  —  4  ac  =  0),  we  have 

aa^  +  6a;  +  c  =  a  (a;  —  7\y. 

Hence  aoi^  -{-bx-hc  has  the  same  sign  as  a  for  all  values 
of  X,  except  x  =  o-^.     For  this  value 

ax^ -\- bx -\- c  =  0. 
If  Vi  and  7*2  are  real  (6^  —  4  ac  >  0)  and  Vi  <  r2 


we  have 


ax^  -{-  bx  -^  c  =  a(x  —  ri)  (x  —  r^. 


Then  when  a  is  positive  aaj^  +  6a;  +  c  is  negative  for  all 
values  of  x  between  i\  and  rg  and  positive  for  all  other  values 
except  1\  and  rg.  When  a  is  negative,  aa^  +  6a;  -f  c  is  positive 
for  all  values  of  x  between  rj  and  r^,  and  negative  for  all  other 
vahies  except  rj  and  rg. 

The  following  table  exhibits  these  results  in  convenient  form  : 


a 

&2  -  4  ac 

ax"^  +  bx-^  c 

+ 
+ 
+ 

0 

+ 

+ 

+ 

+  ,  except  for  x  =  n. 

—  for  values  of  x  between  n  and  r^ ;  +  for  all  other 
values,  except  n  and  Vi. 

-,  except  for  a;  =  n. 

+  for  values  of  x  between  n  and  r^ ;  —  for  all  other 
values,  except  ri  and  Vi. 

Example.  —  For  what  values  of  a:  is  a;^  —  10  >  5  +  a:  —  a;^  ? 

By  Theorem  I  this  inequality  is  equivalent  to 
2  x2  -  X  -  16  >  0. 

Here  n  =—  f,  r<i  =  3,  and  a  =  2.  Hence  the  given  inequality  is  sat- 
isfied by  all  values  of  x  less  than  —  |  and  by  all  values  greater  than  3. 

We  can  treat  quadratic  inequalities  by  completing  the  square,  as  is 
done  in  solving  quadratic  equations,  if  we  observe  proper  precautions.  If 
we  treat  the  preceding  example  in  this  way,  we  get 


INEQUALITIES 


225 


But  we  cannot  conclude  from  this  that 

a;  -  i  >  ±  V-- 
All  we  can  conclude  is  that  either 

and  these  two  inequalities  lead  to  the  results  already  obtained. 

184.    Graphical  interpretation   of  linear   and   quadratic   in- 
equalities. —  The  values   of  x  for   which   aa;  +  6  >  0   are  the 

abscissae  of  the  points  on  the  locus  of  the  equation 

y  =  ax-{-h 

that  are  above  the  a^axis,  and  the 
values  of  x  for  which  aa;  +  6  <  0  are 
the  abscissae  of  the  points  on  this 
locus  that  are  below  the  rc-axis. 
The  value  of  x  for  which  y  =  ax-{-b 
is  the  abscissa  of  the  point  of  inter- 
section of  this  locus  and  the  cc-axis. 

Thus,  a  glance  at  the  locus  of  the  equation 
y  =  Sx-\-6 
tells  us  for  what  values  of  x  the  expression 

3  X  +  5  >  0,  =  0,  or  <  0. 

In  a  similar  way,  an  inspection  of  the 
locus  of  the  equation 

y  =  ax"^  -h  bx  +  c 

tells  us  for  what  values  of  x  the  expression 

ax^  +  6a;  +  c  >  0,  =  0,  or  <  0. 

Suppose  we  wished  to  solve  graphically 
the  inequality  considered  in  the  latter  part 
of  the  preceding  article.  We  draw  the 
locus  of  the  equation  y  =  2x^  —  x—  16  and 
note  the  ahscissse  of  the  points  on  the  locus 
that  are  above  the  a;-axis. 


226 


COLLEGE  ALGEBRA 


A+ 


-V 


EXERCISES 

1.  Does  the  proof  given  in  Theorem  II  of  §  182  cover  the 
case  in  which  both  members  of  the  inequality  are  divided  by 
the  same  positive  number  ? 

2.  Prove  that  a? -\-y^>2xy, 
whenever  x  and  y  are  unequal  real  numbers. 

Hint.  —  This  inequality  is  by  Theorem  I  equivalent  to 

3.  Show  that  x^->2    ^x  -  O"^  >  <? 

X  J       ' 

for  every  positive  value  of  x,  except  1. 

What  is  the  relation  between  the  two  members  of  this  in- 
equality when  a?  =  1  ? 

4.  Prove  that  if  x,  y,  and  z  are  unequal  positive  numbers 

^"^  +  y^  -\- z^  >  ^y  +  y^  ■\-  ^^• 


What  if  X  and  y  are  equal  ? 


5.  Which  is  the  greater,  the  arithmetic  mean,  or  the  posi- 
tive geometric  mean,  of  two  unequal  positive  numbers  ? 

Find  for  what  values  of  x  the  following  inequalities  hold  and 
interpret  your  results  graphically : 

6.  2  a;  — 5  <  a; +  2. 


7. 


8.  Ta;-f 2a;>3ar^  +  5a;  +  l-f4«2_33. 

9.  5a?  +  6<0.  12.   ar^>4. 

13.  (2a;-3)2<9. 

<0. 

14.  a;(a;-2)     =0. 
11.    _4a;2_p20x-25<0.  i>0. 


X 

2 

-1 

3a; 

X 

3a; 

4 

0 

< 

2 

5 

5 

2 

1 

f>0. 

10.   a^  +  2  X  +  5  ]  =  0. 

l<0. 


INEQUALITIES 


227 


15. 


3       2-1 

X       2  X    x-\-l 
—  ^xx  —  2       3x 


> 


r>o. 

16.  (;2x-\-5)(x-2)\  =0. 

[<0. 
f  >1. 

17.  (x-2f 


X     2x        0 
4        7        1 

0    -2    -6 

21.    10-x>2a?. 

x  +  ^ 


22. 


1. 
1<1.  23. 

18.  25a^  +  18a;4-90<0. 

19.  {x-r)(x-2){x-3)>Q.      24.   -A 

20.  x{x-{-lf<0. 


x  +  2 
2a;-l 


3  a;  -  4 


>0. 
<0. 


x  —  b 


>0. 

=  0. 

<o. 


25.  For  what  values  of  m  will  the  line 

y  =  ma;  +  5 

have  no  points  in  common  with  the  circle 

a.«2  -f  2/2  =  9  ? 

26.  For  what  values  of  h  will  the  line 

y  =  ^x-\-b 

have  no  points  in  common  with  the  parabola 

2/2  =  8a;? 

27.  For  what  values  of  h  will  the  loci  of  the  equations 

2/  =  3a;H-6 
and  «2^2/2-2a;  +  4?/  +  l  =  0 

have  one  or  more  points  in  common  ? 


CHAPTER   XVI 
PARTIAL  FRACTIONS 

185.  It  is  explained  in  elementary  algebra  how  to  express 
the  sum  of  several  rational  fractions  as  a  single  fraction  whose 
denominator  is  the  lowest  common  multiple  of  the  denominators 
of  the  given  fractions. 

In  certain  problems  it  becomes  desirable  to  perform  the  in- 
verse operation  on  rational  fractions  of  a  particular  form  — 
namely,  to  express  a  fraction  whose  numerator  and  denomina- 
tor are  polynomials  in  x  as  the  sum  of  two  or  more  fractions 
the  lowest  common  multiple  of  whose  denominators  is  the  de- 
nominator of  the  given  fraction.  These  fractions  whose  sum 
equals  the  given  fraction  are  called  partial  fractions. 

We  shall  consider  in  this  chapter  a  systematic  means  of  de- 
composing rational  fractions  of  the  kind  described  into  partial 
fractions. 

186.  If  the  fraction  is-^^  and  the  degree  off(x)  is  equal  to, 

or  greater  than,  the  degree  of  g{x),  we  can  perform  the  division 
indicated  by  the  fraction.  If  the  resulting  quotient  is  the 
polynomial  Q(x)  and  the  remainder  is  JR{x),  we  have  (§  15) 

y(x)-^^'')  +  g{xy 

Since  the  remainder  is  of  lower  degree  than  the  denominator, 
we  see  that  any  rational  fraction  in  x  whose  numerator  is  of  as 
high  a  degree  as  its  denominator  is  equal  to  the  sum  of  a  poly- 
nomial and  a  fraction  whose  numerator  is  of  lower  degree  than 
its  denominator.  In  discussing  the  decomposition  of  a  rational 
fraction  we  shall  therefore  assume  that  its  numerator  is  of 
lower  degree  than  its  denominator. 

228 


PARTIAL  FRACTIONS  229 

f(x) 

187.  Theorem.  —  Any  such  fraction  '^-j-^  can  he  decomposed 

into  the  sum  of  fractions  of  the  type — ,  tchere  A  is  a  con- 
stant and  (x—  ay  is  a  factor  ofg(x). 

The  proof  of  this  theorem  will  be  omitted.     We  shall  here 

take  it  for  granted  and  consider  the  problem  of  determining 

f  x) 
these  partial  fractions  for  a  given  fraction  ^—^. 

188.  Case  I. —  When  the  linear  factors  ofg(x)  are  distinct. 
Example.  —  Decompose  ^-^t into  partial  fractions. 

X    —  O  X  •f"  o 

The  numerator  is  of  lower  degree  than  the  denominator,  and  the  Hnear 
factors  of  the  denominator  are  x  —  2  and  x  —  3.  We  know  therefore  from 
the  theorem  nf  §  187,  that 

2  a; +  3      ^    A  B  ,^. 

x^-bx  +  Q     x-2     x-Z'  ^  ^ 

where  A  and  B  are  certain  constants. 

The  problem  is  to  find  the  values  of  these  constants. 
Clearing  (1)  of  fractions,  we  get 

2x  +  ^  =  A{x-^)-\-B{x-2).  (2) 

Now  the  two  members  (1)  must  be  equal  for  all  values  of  x  except  2 

and  3  (for  these  exceptional  values  of  x  neither  member  of  (1)  has  any 

value),  and  therefore  the  two  members  of  (2)  which  are  polynomials  of 

the  first  degree  are  equal  for  all  these  values.    Hence  by  §  149,  Cor.,  the 

coefficients  of  the  like  powers  of  x  in  the  two  members  of  (2)  must  be 

equal.     That  is 

A+B  =  2, 

and  -3^-25  =  3. 

Solving  these  two  equations  for  A  and  B,  we  get 

A  =  -1  and  B  =  d. 
Hence  2a:4-3      ^^-7_^     9 


x2-5a;4-6     x-2     x-3 
The  student  should  verify  that  this  is  correct. 


230  COLLEGE  ALGEBRA 

If,  in  general,  g{x)  is  the  product  of  n  distinct  linear  factors, 
or  g(x)  =  (a^x  +  b{)  (a^x  +  h^  -"  {a^x  +  6 J, 

then,  by  §  187, 

g{x)      a^x  +  h^      a^x  +  h^  a^a;  +  6„* 

Clearing  of  fractions,  we  get 
f{x)  =  A^{a^x-^h^{a^x  +  63)  •-.  (a„a;  +  6„) 

+  A^{a^x 4- h^) {a^x  +  \)  '■•  (a^x  +  b„) 

+  •••  +AXaiX-}-b{)(a^+b2)  •••(a„_ia;+&„-i).  (2) 

Now  the  two  members  of  (1)  must  be  equal  for  all  values  of 

X  except i, -,  '"  J -]  and  therefore  (2)  must  be  true 

for  all  these  values.  Moreover  the  left  member  of  (2)  is  of 
degree  not  greater  than  n  —  1  and  the  right  member  is  of 
.degree  n  —  1.  Then  by  (1149,  Cor.,  the  coefficients  of  like 
powers  of  x  in  these  two  members  must  be  equal.  Hence  the 
n  unknowns,  A^,  A2,  '•' ,  A^  must  satisfy  the  n linear  equations 
obtained  by  equating  the  coefficients  of  these  like  powers.  By 
virtue  of  the  theorem  of  §  187  we  know  that  these  equations 
cannot  be  inconsistent,  and  that  therefore  they  can  be  solved. 

When  we  have  solved  these  equations  and  substituted  the 
resulting  values  of  Ai^  A^y  -",  A^  in  (1)  we  have  the  desired 

decomposition  of  -^^^  into  partial  fractions. 

EXERCISES  * 


•^7. 


Decompose  the  following  fractions  into  partial  fractions : 

2x  g  2a;-l 

x^-l  '    12aj3  +  20a^  +  3a;' 


4  a;2  -  2  a;  +  3 ^     2  a^- 5ar^  +  a;  + 7 

(a;-l)(5a;-2)(2x+3)*  "         x'-5x''  +  4: 


PARTIAL  FRACTIONS  231 

^+^+1  ^  ^  +  ^3  0^  +  36 . 

g  3  0^=^ 3a;-4 

5 15.  11^-^ 


(2x-\-l){x^2){x-^3) 

^O      8. 

2x  +  5 

(x^-9)(7x-\-2) 

1     O      9. 

2a^-\-x'-x-{-l 

(x-l)(x^^3x-\-2) 

1  ^       10. 

3x-2 

(a^  +  a;-2)(a;  +  3; 

11 

2x 

^-x-2 

12 

4 

30^2  +  11  a;- 20 


16. 


17. 


18. 


19. 


20. 


oi?--\-  \x-\-h 


x-\-l 
(x  —  a)(x—  b) 

x^-h2x-\-3 
(2a;-5)(5a;-2)* 

(x-{-3)(x-4) 
{x-l)(x-\-2)(x-5) 

3x4-5 


(x-j-lXx" -  6 a;  +  5)  (4.x  +  7)(2 aj  +  3) 

189.   Case  II.  —  When  the  linear  factors  of  g(x)  are  not  all 

distinct. 

Q  /W.2  _L  2  a; 4 

Example.  —  Decompose  — into  partial  fractions. 

^        (x2  +  6a;  +  9)(x-l) 

The  numerator  is  of  lower  degree  than  the  denominator,  and  the 
factors  of  the  denominator  are  (x  +  3)2  and  x  —  1.  We  know  therefore 
from  the  theorem  of  §  187  that 

3x2  4-2x-4        ^     A  B  C  .^. 

(x2  +  6x  +  9)(x-l)      x-1      (x  +  3)2     x  +  3' 

since  the  three  fractions  in  the  right  member  represent  all  the  types  of 
partial  fractions  mentioned  in  the  theorem. 
Clearing  of  fractions,  we  get 

3x2  4-2x-4  =  A{x  +  3)2  -f  B{x  -  1)+  C(x  -  1)  (x  +  3).        (2) 

By  an  argument  similar  to  the  one  used  in  Case  I  we  see  that  the  coeffi- 
cients of  like  powers  of  x  in  the  two  members  of   (2)  must  be  equal. 


^ 


232  COLLEGE  ALGEBRA 

Hence  3  =  ^+0', 

-^  =  9A-B-SC; 

Thisgives   . 3x^  +  2a^-4        ^_^ ^_+_iL 

(x2  +  6a;  +  9)(a;-l)      a;  -  1      (x  +  3)2^a;  +  3 

as  the  desired  decomposition. 

If,  in  general, 

g  (x)  =  (a^x  +  &i)«x  (aga;  +  b^y^  •  •  •  (a,cc  +  hT', 
where  ?ij  -j.  ^2  +  •  •  •  +  n^  =  7i, 

then  by  §  187,  we  have 

/(^)  = A  A,  _         A, 


(ajjo;  +  ftg)"*     (as^J  +  &2)"2"^  ^2^  +  &2 


+ ^ + ^ +...  4-_^,       (1) 

(a^  +  &*)"*     (a^a;  +  ^x:)*^*"^  %aJ+ &* 

since  the  fractions  in  the  right  member  represent  all  the  types 
of  partial  fractions  mentioned  in  the  theorem. 

If  we  clear  (1)  of  fractions,  the  left  member  of  the  resulting 
equation  will  be  of  degree  less  than  n  and  the  right  member 
will  be  of  degree  n  —  1.  Therefore  when  we  equate  the 
coefficients  of  like  powers  of  x  in  this  equation  in  accordance 
with  the  principles  already  explained  we  shall  get  n  linear 
equations  in  the  n  unknowns 

Ai,  A^  •••,  An^\  Bi,  B2,  •••,  Bn^;  •••;  Zj,  X2,  •••,  Ln*- 


PARTIAL  FRACTIONS  233 

Moreover  these  equations  must  be  solvable,  since  if  they 
were  not,  the  decomposition  mentioned  in  the  theorem  of 
§  187  would  be  impossible. 

If  therefore  we  solve  them  and  substitute  the  values  of  the 
unknowns  in  (1),  we  shall  have  the  desired  decomposition  of 

q(x) 

^^  ^  EXERCISES 

Decompose  the  following  fractions  into  partial  fractions : 
^         3x  +  l 


2.      ^-1 


x{x-\-V) 


2a^-f  3a;-l  *   Q(? -\- ^  x" -^  27  x  +  21 


14.  ^^  +  2 


^  '^\   "         (ar'_l)(a:  +  l) 

__  ^7^+36^+27 

'"    ^''Mih'  (2.  +  3)X3.+  2) 
2                                           16.  ^  +  2 


a^  +  a^-fa?4-2  17.    ^ 

(;4a^  +  12»  +  9)(a;+l/  (a:  +  l)>-l)3 

-    (2  a; +  6)'  18.   i^±3 

g           3ar'-5o0  jg  ^  +  3 


(3  a; -5)^(2 a; +  1)  (a^-9)(a!-3) 

10.    ,^±1    .  20.  6 


(a^-4)2  (a;^  +  3a;H-2)' 


234  COLLEGE  ALGEBRA 

190.   If  we  apply  the  preceding  methods  to  the  decomposi- 

tion  of  ;  ,  we  get 

(a^  +  l)(a;  +  d) 

20a;-f-40      _l  +  7i      l-7i         2 


(a^  +  l)(a;  +  3)        x-\-i         x  —  i       a; +3 

The  appearance  of  imaginary  coefficients  in  this  decomposi- 
tion can  be  avoided  by  not  carrying  the  decomposition  so  far. 
Thus  if  we  retrace  our  steps  in  part  by  expressing  the  sum  of 
the  first  two  fractions  of  this  decomposition  as  a  single  fraction, 

^^^^^  200^  +  40      ^20^  +  14         2 

(a^  +  l)(a;  +  3)        a;2  +  l        a;  +  3* 

Here  there  are  no  imaginary  coefficients  in  the  final  form 
and  the  denominators  are  of  the  first  or  second  degree. 

If  the  coefficients  of  the  original  fraction  are  real,  the 
decomposition  can  always  be  stopped  at  a  point  at  which 
the  coefficients  are  real  and  the  denominators  are  of  the  first 
or  the  second  degree  or  powers  of  such  expressions. 

By  §  148,  we  know  that  any  polynomial,  as  g  {x),  of  degree  n 
is  the  product  of  n  factors.  Moreover  if  the  coefficients  of  g  {x) 
are  real,  the  imaginary  factors  (that  is,  those  with  imaginary 
coefficients)  occur  in  conjugate  pairs  (§  151),  if  at  all.  The 
part  of  the  decomposition  that  corresponds  to  such  a  pair  of 
imaginary  factors  of  the  first  degree  will  be  of  the  form 

A  B 

x  —  a  —  ih     x  —  a-\-ib 

But  this  is  equal  to 

(A  +  B)x-^(-Aa-Ba-\-  iAb  -  iBb) 
3f-2ax-\-a^-]-b^ 

Then  in  the  decomposition  we  can  put  the  single  fraction 
AiX  4-  Bi 
x^-2ax  +  a^-\-  W' 
where  A^=A-\-B  and  B^=  —  Aa  —  Ba-{-  iAb  —  iBb. 


PARTIAL  FRACTIONS  235 

It  can  be  shown  in  general  that  if  n^  is  the  highest  power  of 
x  —  a  —  ib  and  of  x  —  a  +  ib  by  which  the  denominator  is  divis- 
ible, then  the  sum  of  the  partial  fractions  whose  denominators 
are  the  different  powers  oi  x  —  a  —  ib  and  of  x  — a -\-ib  is  equal 
to  the  sum  of  fractions  of  the  form 

^  A^x-hB^^  A„^_,x  4-  ^n,-i  .. 

(a;2  -2ax-\-  a^+  b^y^'    (x"  -  2  aa;  +  a^  +  62)n,-i'  '"> 


A,x  -f  B, 


x^-2ax  +  a'^-\-b^' 

where  the  ^4's  and  ^'s  do  not  contain  x. 

Then  in  the  original  decomposition  in  place  of  a  pair  of  frac- 
tions whose  denominators  are  conjugate  imaginaries  we  can 
place  a  single  fraction  whose  numerator  is  of  the  first  degree 
with  undetermined  coefficients  and  whose  denominator  is  a  real 
factor  of  g  (x)  of  the  second  degree  or  a  power  of  such  a  factor. 
The  resulting  equations  for  determining  the  undetermined  con- 
stants in  the  numerators  of  the  partial  fractions  will  have  only 
real  coefficients  and  will  be  of  the  first  degree.  .  Hence  these 
undetermined  constants  must  have  real  values. 

This  partial  decomposition  of  a  rational  fraction  has  there- 
fore the  advantage  of  involving  only  real  coefficients,  and  it  is 
sufficient  for  the  uses  to  which  decomposition  into  partial 
fractions  is  put. 

Example  1. — Decompose  ^-i into  real  partial  fractions. 

(x2 +!)(:*: +  3) 

Put  20  a;  +  40       _  Ax -\-  B         C 

(x2  4-l)(a;  +  3)       x^ -^  1       x  +  s" 

Clearing  of  fractions  we  get 

20  x  +  40  =  {Ax  +  5)(x  +  3)  +  0(^2  +  1). 

Equating  coeflBcients  of  like  powers  of  x, 
0  =  A+C, 
20  =  SA-\-B, 
40  =  35  +  0. 


236  COLLEGE  ALGEBRA 

Whence  A  =  2,  B  =  li,  C  =  —  2,  and  the  decomposition  is 

20  X  +  40        ^  2  a;  +  14  2 

(a;2+l)(x  +  3)        x^+1       x  +  s' 

Q  y K 

Example  2. —  Decompose into  real  partial  fractions. 

(x2  +  a:  +  l)2(x  +  1) 

3x-5  Ax-\-B       ,     Cx+  D     ,     E 


{X^  +  X  +  lf{X  +1)         (iC2  +  X  +  1)'-^       x^  +  X+1        X  +  1 

Clearing  of  fractions, 

3x-5  =  (^x  +  -B)(x+  l)  +  (Cx  +  i>)(x2+a;+  l)(x+l)+^(x2+x+l)2. 

Equating  the  coefficients  of  like  powers  of  x, 

G  +  E=0, 

2C+D  +  2E  =  0, 

A-\-2C+2D-\-SE  =  0, 

A-\-B-^C+2D  +  2E  =  S, 

B  +  D-\-E  =  -5. 

Whence  J.  =  8,  5  =  3,  0=8,  Z)=0,  E=-S;  and  the  decomposition  is 

3x-5  ^       8x  +  3  8x 8_ 

(x2  +  x  + l)2(x+ 1)      (x2  +  x  +  l)2     a:2  +  x+l     x  +  l' 

EXERCISES 

Decompose  the  following  fractions   into   real  partial  frac- 
^tions : 

6.  '^+^ 


'  1. 

2x^-5 

^    2. 

Zx-2 

0     *K 

a^_5aj2  +  2a;  +  l 

a^  +  5a;2 -1-4 

d  4. 

a^  +  a;2  +  a? 

(a;2  +  2aj  +  5)(a;-l)2 


4  a;2  -{-  a;  —  6 
6 


8. 


(05^+ 1)^ 


PARTIAL  FRA.CTIONS  237 

11  a^  4- 11  15         x'  +  x  +  2 

(2a;  +  3)(»2  +  x-f-3)'  *  ^^x'^^x  +  l 

,^    a;2  +  2a;  +  3  ,.  4a^4-4a;2  4- 8  cc  +  6 

lU.     •  Id.  — • 

a?"-!  x<+2a;2 

11    ^±^-  "•  T^- 


3a;-l  18.    -2x^-9a;-10 

/^  -f  1)2*  cc^  +  6  a;2  +  5  a; 


13. 


4a;H-5  .o        2a^4-14 


2/4-81  a^-2ic3  +  l 


CHAPTER   XVII 

LOGARITHMS 

191.   Definition   of  irrational   exponents.  —  In   §   58  it  was 

stated  that  irrational  exponents  could  be  used  in  a  way  con- 
sistent with  the  five  fundamental  laws  of  exponents.  This 
statement  is  based  upon  the  meaning  we  attach  to  these 
exponents.  We  cannot  give  a  complete  justification  of  it 
here,  but  shall  indicate  briefly  what  this  meaning  is. 

In  the  first  place  the  student  should  recall  that  a  decimal 
fraction  can  always  be  written  as  a  common  fraction,  and  that 
accordingly  a  number  with  a  decimal  exponent  can  be  ex- 
pressed in  a  way  with  which  he  is  familiar.     For  example, 

Now,  as  he  can  readily  verify,  the  numbers 
1,  1.4,  1.41,  1.414,  1.4142,   ..., 

which  are  obtained  in  the  process  of  finding  the  approximate 
square  root  of  2,  are  such  that  their  successive  squares  are 
closer  and  closer  to  2.  As  a  matter  of  fact,  it  can  be  shown 
that  when  this  sequence  is  continued  indefinitely  the  successive 
numbers  approach  a  certain  definite  limit,  which  is  the  positive 
square  root  of  2.     (See  §  202  for  the  definition  of  a  limit.) 

It  can  be  shown  further  that  the  numbers  of  the  corre- 
sponding sequence 

a\  a}\  a'*\  a}'^\  a^-^^V  •••, 

also  approach  a  limit.  It  is  this  limit  that  we  denote  by  the 
symbol  a^~^. 

238 


LOGARITHMS 


231 


In  general,  every   irrational  number  n   is  the   limit  of  a 
sequence  of  rational  numbers 

Wu  ^2,  %,   .••, 

and  the  numbers  of  the  sequence 

a**!,  a**!,  a**3,  ... 
approach  a  limit,  which  we  indicate  by  the  symbol  a^ 

192.   Definition.     If  a>0   and  a'  =  h,  x  is  said  to  be  the 
logarithm  of  b,  and  b  the  antilogarithm  of  x,  to  the  base  a. 

The  symbol  for  the  logarithm  of  b  to  the  base  a  is  log^  b. 

Thus,  25  =  32,  and  therefore  5  is  the  logarithm  of  32,  and  32  the  anti- 
logarithm  of  5,  to  the  base  2,  or  5  =  log2  32. 


EXERCISES 

1.  log39=  ?   log93=  ?    log6l=  ?  log,a=  ?   \og,a^=  ? 

2.  Complete  the  following  table  : 


^ 


Number 

Base 

Logarithm 

243 

3 

2 

8 

10 

3 

5 

4 

2 

16 

343 

3 

3 

i 

3 

A1 


^. 


^ 


'.^ 


^^^.'> 


193.   We   can  take   for   the   base   a  any   positive   number 
except  1. 

Theorem.  —  For  any  positive  number  b  and  any  positive  numr 
ber  a,  except  1,  there  is  a  real  number  x  such  that 

a^^b. 


240  COLLEGE  ALGEBRA 

In  other  words,  every  positive  number  has  a  real  logarithm 
with  respect  to  any  positive  base  except.  1. 

The  proof  of  this  theorem  is  too  difficult  for  this  book,  and 
tjLfi/efore  is  omitted. 

Negative  numbers  have  no  real  logarithms.  It  can  be  shown 
that  the  logarithm  of  a  negative  number  to  a  positive  base  is 
imaginary.  We  shall  confine  our  attention  here  to  real 
jogarithms. 

194.  The  logarithms  of  all  positive  numbers  with  respect  to 
any  given  positive  base  different  from  1  form  a  system  of  real 
numbers  that  possess  the  following  properties  : 

I.  Tlie  logarithm  of  the  product  of  two  or  more  numbers  is 
equal  to  the  sum  of  the  logarithms  of  these  numbers. 

For,  if  log«  b  =  x,  log«  c  =  y, 

or  a'  —  b  and  a^  =  c, 

then  by  multiplication 

a'^+y  =  be. 

But  this  is  only  another  way  of  saying 

.     log^  6c  =  a;  +  2/  =  ^^^a  &  +  log„  c. 

This  proves  the  theorem  as  far  as  concerns  the  product  of 
two  numbers. 

The  completion  of  the  proof  is  left  to  the  student. 

II.  Tlie  logarithm  of  the  quotient  of  two  numbers  is  equal  to 
the  logarithm  of  the  dividend  minus  the  logarithm  of  the  divisor. 

For,  if  log„  b  =  x,  log«  c  =  y, 

or  a'  =b,  a^  =  c, 

then  by  division,     a*""  =  -  • 

c 

But  this  is  only  another  way  of  saying 

log« -  =  x-y  =  log^ b  -  loga c. 
c 


LOGARITHMS  241 

III.   If  m  is  any  real  number,  the  logarithm  of  b"^  is  equal  to  m 
times  the  logarithm  of  b. 

For,  if  loga  b  =  x, 

or  a^  =  by 

then  we  have,  by  equating  the  mth  powers  of  the  two  members 
of  this  equation,       ^«.^^„^ 

Hence  log^  b'^  =  mx  =  m  » log^  b. 

If  in  III  m  is  equal  to  -,  where  n  is  an  integer,  we  have, 

n 

the  logarithm  of  the  positive  nth  root  of  a  positive  number  is  equal 

to  the  logarithm  of  the  number  divided  by  n. 

EXERCISES 

Given  log^  2  =  .3010,         logio  3  =  .4771,         logio  7  =  .8451 
logg  7  =  1.7712,       logg  11  =  2.1827;  find: 
1.   logio  14. 
Hint.  — 14  =  2  •  7.     Hence  logio  14  =  logio  2  +  logio  7. 


2.  logaV^.  7.  logio V2«.7^ 

3.  logio^v^.  8.  logio  2058. 

4.  logsJ^^  9.  log3V7. 

5.  logio V2i.  10.  loga^^TT. 

6.  log3^5^5929. 

195.   The  systems  of  logarithms  most  frequently  used. — 

For  reasons  that  are  explained  in  the  Differential  Calculus  it  is 
convenient  for  certain  purposes  to  use  as  base  of  the  system  of 
logarithms  an  irrational  number  whose  approximate  value  is 
2.71828.  Logarithms  to  this  base  are  called  natural  logarithms. 
On  the  other  hand,  for  purposes  of  numerical  computation, 
there  is  an  advantage  in  using  the  base  10.  The  reason  for 
this  will  be  explained  in  §  198,  I.  Logarithms  to  the  base  10 
are  called  common  logarithms. 


242  COLLEGE  ALGEBRA 

196.  Change  of  base. — Since  different  systems  of  logarithms 
are  in  use,  it  is  important  to  know  how  to  change  from  one 
system  to  another.  The  following  theorem  explains  how  this 
can  be  done. 

Theorem.  —  The  logarithm  of  a  number  to  the  base  b  is  equal 

to  the  product  of  its  logarithm  to  the  base  a  and  the  logarithm  of  a 
to  the  base  b. 

If  log„ X  =  u  and  logj  x  =  v',  that  is, 
if  a''=x 

and  b"  =  x, 

then  a"  =  5". 

Hence  a  =  b'^, 

or  -  =  logj  a. 

u 

Hence  v  =  U'  logj  a, 

or  logj  a;  =  log«  aj  •  logj  a. 

This  proves  the  theorem. 

The  following  theorem  enables  us  to  express  this  relation  in 
another  form,  which  is  sometimes  useful. 

Theorem.  —  The  logarithm  of  a  to  the  base  b  is  the  reciprocal  of 
the  logarithm  of  b  to  the  base  a. 

For,  if  logj,  a  =  x,  or  a  =  6* ; 

I  1 

then  a""  —  b.  or  log„  6  =  - . 

X 

Hence  log.  a  = :; • 

log«6 

The  relation  proved  in  the   preceding  theorem  can   now  be 
written  thus:  1 

197.  Common  logarithms.  —  We  shall  confine  our  atten- 
tion in  the  rest  of  this  chapter  to  common  logarithms  and 
it  will  therefore  be  unnecessary  to  indicate  the  base  in  the 


LOGARITHMS  243 

symbol  for  a  logarithm.  Thus  we  shall  write  log  2  instead  of 
logio  2. 

Now  it  is  easy  to  see  what  the  logarithm  of  100,  or  of  1000,  is ; 
but  it  is  not  easy  to  determine  the  logarithms  of  most  numbers, 
say  20,  for  example.  The  approximate  value  of  the  logarithm 
of  such  a  number  (that  is,  of  a  number  that  is  not  an  integral 
power  of  10)  can  be  determined  only  by  a  series  of  computa- 
tions that  are  too  long  and  complicated  to  be  explained  here. 

The  integral  part  of  a  logarithm  is  called  the  characteristic 
of  the  logarithm  and  can  be  determined  by  inspection  in  a  way 
that  we  shall  explain  (§  198).  The  fractional  part  of  the  log- 
arithm is  called  the  mantissa  and  is  given  in  a  table.  Thus, 
the  approximate  value  of  the  logarithm  of  20  being  1.3010, 
the  characteristic  is  1  and  the  mantissa  .3010. 

By  virtue  of  their  properties  given  in  §  194,  logarithms  are 
extremely  useful  in  shortening  numerical  computation,  and  it 
is  for  this  reason  we  are  considering  them  here. 

Suppose,  for  example,  we  want  the  fifth  root  of  20.  We  have  already 
seen  that  j^ggo  =  1.3010. 

But  by  §  194,  III,  log  v/20  =  ^  log  20. 

Hence  log  v^20  =  .2602. 

Now  the  number  whose  logarithm  is  .2602  can  be  determined  approxi- 
mately from  the  table  in  a  way  to  be  explained  in  §  198,  II.  And  when 
this  is  determined  we  shall  have  the  approximate  value  of  \/20. 

The  student  would  appreciate  the  advantage  in  this  use  of  loga- 
rithms if  he  were  to  attempt  to  find  the  fifth  root  of  20  directly. 

The  operations  of  multiplication,  division,  and  raising  to 
powers  can  also  often  be  shortened  by  the  use  of  logarithms. 

But  for  this  numerical  work  it  is  necessary  to  use  a  table 
of  logarithms. 

198.  Use  of  a  logarithmic  table.  —  We  proceed  therefore 
to  explain  the  two  ways  in  which  the  table  can  be  used.  The 
procedure  is  based  upon  the  fact  that  if  a  and  b  are  greater 
than  1  and  b  is  greater  than  a  then  log  b  >  log  a. 


244  COLLEGE  ALGEBRA- 

I.  To  find  the  logarithm  of  a  number.  —  The  table  in  this 
book  is  so  arranged  that  the  mantissa  of  the  logarithm  of  a  num- 
ber of  three  digits  can  be  taken  out  directly.  To  do  this  we 
look  in  the  first  column  for  the  first  two  digits  (counting  from 
the  left)  and  then  over  on  this  row  to  the  column  headed  by 
the  right-hand  digit.  The  number  in  this  row  and  column  is 
the  mantissa  we  are  looking  for.  It  is  understood  that  all  the 
numbers  in  the  body  of  the  table  are  to  be  preceded  by  the 
decimal  point.  Thus,  the  mantissa  of  the  logarithm  of  483  is 
.6839. 

The  characteristic  is  determined  without  the  use  of  the 
table  from  the  following  considerations: 

10«  =  1,  orlogl  =  0. 
10^  =  10,  or  log  10  =  1. 

102=100,  or  log  100  =  2. 
103  =  1000,  or  log  1000  =3. 


Hence  the  logarithm  of  any  number  between  1  and  10  lies 
between  0  and  1,  and  has  therefore  the  characteristic  0. 
Moreover  every  such  number  (when  written  as  an  integer  or 
a  decimal)  has  one  digit  to  the  left  of  the  decimal  point.  The 
logarithm  of  any  number  between  10  and  100  is  between  1 
and  2  and  has  the  characteristic  1.  Such  a  number  has  two 
digits  to  the  left  of  the  decimal  point. 

In  general,  since  10**  is  the  smallest  integer  with  n-\-l  digits 
the  characteristic  of  the  logarithm  of  a  number  greater  than  1  is 
one  less  than  the  number  of  digits  to  the  left  of  the  decimal  point 
in  the  number. 

We  agree  to  say  that  the  numbers  2015,  201,500,  2.015,  and 
.0002015  have  the  same  sequence  of  digits.  In  general,  two 
numbers  that  have  the  same  digits  in  the  same  order  and  dif- 
fer only  in  the  position  of  the  decimal  point  and  in  the 
ciphers  that  may  be  necessary  to  indicate  the  position  of  the 
.decimal  point  are  said  to  have  the  same  sequence  of  digits. 


LOGARITHMS  245 

The  position  of  the  decimal  point  in  a  number  can  be 
changed  any  number  of  places  to  the  right  or  left  by  multi- 
plying or  dividing  the  number  by  some  integral  power  of  10. 
Moreover  the  logarithm  of  any  integral  power  of  10  is  an 
integer. 

For  example,  log  432  =  2.6355  ; 

that  is,  432  =  1026355. 

Dividing  each  member  of  this  equation  by  10,  we  get 

43.2  =  101-6355. 

Hence,  log  43.2  =  1.6355. 

In  a  similar  way  we  see  that 

log  4.32  =  0.6355, 
log  .432  =  .6355-1, 
log  .0432  =  .6355 -2, 
log  .00432  =  .6355 -3, 


From  this  we  see  that,  for  example, 

log  .432  =  .6355  -  1  =  -  .3645. 

But  a  logarithm  is  usually  not  written  in  this  way,  but  thus, 
9.6355  - 10. 

Similarly,  we  say 

log  .0432  =  8.6355 -10, 
log  .00432  =  7.6355  -  10. 

This  agreement  always  to  write  the  logarithm  in  such  a  way 
as  to  have  its  mantissa  positive  is  based  on  grounds  of  con- 
venience in  the  use  of  the  tables. 

If  there  are  n  ciphers  immediately  following  the  decimal 
point  in  a  number  less  than  1,  the  characteristic  will  be  taken 
as  —  n  —  1.     It  is  convenient  to  write  this  9  —  71  —  10, 


246  COLLEGE  ALGEBRA 

If  we  agree  to  write  the  logarithms  of  numbers  less  than 
1  in  the  way  indicated  here,  we  can  make  the  following 
statements : 

The  mantissa  of  the  logarithm  of  a  number  is  always  positive. 

Tlie  logarithms  of  two  numbers  that  have  the  same  sequence  of 
digits  have  the  same  mantissa  and  differ  only  in  their  charac- 
teristics. 

The  characteristic  of  the  logarithm  of  a  number  less  than  1  is 
equal  to  9  —  w  — 10,  where  n  is  the  number  of  ciphers  immedi- 
ately following  the  decimal  point  in  the  number.  This  character- 
istic is  written  in  two  parts.  Tlie  first  part,  9  —  7i,  is  written 
at  the  left  of  the  mantissa  and  the  —  10  at  the  right. 

Thus  log  .00432  =  7.6355  -  10. 

Here  w  =  2  and  9  —  w  =  7. 

It  is  clear  from  what  has  been  said  that  in  looking  in  the 
table  for  the  mantissa  of  the  logarithm  of  a  number  we  do  not 
need  to  pay  any  attention  to  the  position  of  the  decimal  point 
in  the  number,  —  the  characteristic  is  the  only  thing  about 
the  logarithm  that  is  affected  by  the  position  of  the  decimal 
point. 

The  finding  of  the  logarithm  of  a  number  of  more  than  three 
digits  from  this  table  is  not  so  simple.  The  method  of  pro- 
cedure is  best  illustrated  by  an  example. 

Find  log  5421. 

From  the  tahle  log  5420  =  3. 7340, 

log  5430  =  3.7348. 

These  two  logarithms  differ  by  .0008  and  correspond  to  numbers  that 
differ  by  10.  Now  the  numbers  5420  and  5421  differ  by  1  and  we  assume 
that  the  difference  in  their  logarithms  is  .1  of  the  difference  in  the  loga- 
rithms of  5420  and  6430  ;  that  is,  .1  of  .0008.     This  is  .00008.     Hence 

log  5421  =  3.7340  +  .00008  =  3.7341. 

If  we  were  looking  for  the  logarithm  of  5427,  we  should  add 
.7  of  .0008  to  the  logarithm  of  5420  since  the  difference  be- 


LOGARITHMS  247 

tween  5427  and  5420  is  .7  of  the  difference  between  5430 
and  5420. 

In  doing  this  work  it  is  customary,  in  the  interests  of 
brevity,  to  omit  the  decimal  point  in  giving  the  difference 
between  the  logarithms  of  two  numbers.  Thus,  in  the  ex- 
ample just  given,  we  say  that  the  difference  between  the 
logarithms  of  5420  and  5430  is  8,  instead  of  .0008. 

We  have  assumed  that  the  difference  between  the  loga- 
rithms of  two  numbers  is  proportional  to  the  difference 
between  the  numbers.  This  is  not  strictly  true.  How- 
ever, when  the  difference  between  the  numbers  is  small, 
as  it  is  in  these  cases,  the  error  due  to  this  assumption 
is  very  slight  and  can  safely  be  neglected,  since  most  of 
the  logarithms  given  in  the  table  are  themselves  only 
approximations. 

In  using  foar-place  tables  the  student  should  keep  the  anti- 
logarithms  in  all  his  calculations  down  to  four  significant  fig- 
ures. The  point  of  this  remark  is  illustrated  by  the  fact  that 
43.526  and  43.53  have  the  same  four-place  logarithm. 

From  the  table  log  43. 600  =  1.6395 

log43.500  =  1.6385 

Difference  in  logs.  =        10 

The  difference  between  43.600  and  43.526  is  .26  of  the  difference  be- 
tween 43.500  and  43.600,  and  .26  of  the  difference  between  the  logarithms 
of  these  last  two  numbers  is  3.    Hence 

log  43.526  =  1.6388. 

On  the  other  hand  the  difference  between  43.500  and  43.53  is  .3  of  the 
difference  between  43.500  and  43.600,  and  .3  of  the  difference  between  the 
logarithms  of  these  numbers  is  3,     Hence 

log  43.53  =  1.6388. 

When  the  digits  to  the  right  of  the  fourth  place  (counting  from  the 
left)  are  dropped  off,  increase  the  digit  in  the  fourth  place  by  1  if  the  digit 
in  the  fifth  place  is  greater  than  5,  or  if  the  fourth  digit  is  odd  and  the 
fifth  one  is  equal  to  5  ;  in  other  cases  leave  the  fourth  one  unchanged. 


248  COLLEGE  ALGEBRA 

EXERCISES 
Find  the  logarithms  of  the  following  numbers : 

1.  461.  8.   24.86.  15.  3. 

2.  .0024.  9.   2562.  16.  12. 

3.  500000.  10.   14380.  17.  58.64. 

4.  .000005.  11.   .06473.  18.  5437. 

5.  1934.  12.   374.6.  19.  .6892. 

6.  7.832.  13.   8001.  20.  40.15. 

7.  .6294.  14.   19.03. 

II.  To  find  the  antilogarithm  of  a  logarithm.  —  If  the  man- 
tissa of  the  given  logarithm  is  found  in  the  table,  it  is  an  easy 
matter  to  find  the  antilogarithm.  Suppose,  for  example,  we 
wish  to  find  the  antilogarithm  of  7.9258  — 10. 

Now  .9258  appears  in  the  table  as  the  mantissa  of  the  log- 
arithms of  those  numbers  whose  sequence  of  digits  is  843. 
Since  the  characteristic  is  7  —  10,  the  antilogarithm  must  be  less 
than  1  and  there  must  be  two  ciphers  immediately  following 
the  decimal  point. 

Hence         antilog  (7.9258  - 10)  =  .00843, 
or  log  .00843  =  7.9258  - 10. 

When  the  given  mantissa  is  not  found  in  the  table  the 
matter  is  not  so  simple.  The  method  of  procedure  in  such 
cases  is  illustrated  by  the  following  example : 

Find  antilog  2.6959. 

In  this  part  of  the  tables  when  a  four-place  number  is  in- 
creased by  10,  the  mantissa  of  its  logarithm  is  increased  by  9. 
Hence  we*  assume  that  if  a  mantissa  is  increased  by  1,  the 
number  it  corresponds  to  is  increased  by  ^'.  Since  the  man- 
tissa 6959  is  obtained  by  increasing  the  mantissa  6955  by  4,  the 
antilogarithm  of  6959  should  be  that  of  6955  increased  by 
4x^  =  4.  Hence  the  given  mantissa  must  correspond  to 
numbers  whose  sequence  of  digits  is  4964.     Since  the  charac- 


LOGARITHMS  249 

teristic  is  2,  the  antilogarithm  must  be  greater  than  1,  and  it 
must  have  three  digits  to  the  left  of  the  decimal  point. 

Hence  antilog  2.6959  =  496.4, 

or  log  496.4  =  2.6959. 

The  difference  between  the  mantissae  of  the  logarithms  of 
two  consecutive  numbers  is  called  a  tabular  difference. 

The  preceding  example  suggests  the  following  rule  for  find- 
ing the  antilogarithm  of  a  logarithm  whose  mantissa  does  not 
appear  in  the  table. 

Rule.  —  Find  two  consecutive  mantissce  in  the  table  between 
which  the  given  mantissa  lies,  and  get  the  difference  between  them. 
Multiply  the  difference  between  the  smaller  of  these  mantissce  and 
the  given  mantissa  by  10  and  divide  the  product  by  the  tabular 
difference  just  found.  Annex  the  quotient,  expressed  in  the  nearest 
integer,  to  the  sequence  of  three  digits  corresponding  to  the  smaller 
mantissa  of  the  table. 

In  the  resulting  sequence  of  digits  place  the  decimal  point  as 
indicated  by  the  given  characteristic. 

EXERCISES 
Find  the  antilogarithms  of  the  following : 

1.  2.65^2.  11.  2.0312. 

2.  9.9805-10.  12.  3. 

3.  1.8457.  13.  8. 

4.  8.1644-10.  14.  .8354. 

5.  7.8162-10.  15.  .1870. 

6.  1.6245.  16.  1.8125. 

7.  .1287.  17.  9.8449-10. 

8.  9.1287-10.  18.  7.3950-10. 

9.  1.1287.  19.  2.6045. 

10.   8.9970-10.  20.   8.4857-10. 


250 


COLLEGE  ALGEBRA 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

lO 

0000 

0043 

cx)86 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

II 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

13 

"39 

1173 

1206 

1239 

1271 

1303 

1335 

13^7 

1399 

1430 

14 

1461 

1492 

1523 

^553 

1584 

1614 

1644 

1673 

^703 

1732 

15 

1 761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

371^ 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

32 

5051 

5065 

5079 

5092 

5105 

5"9 

5132 

5145 

5159 

5172 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

35 

5441 

5453 

5465 

5478 

5490 

5502 

55H 

5527 

5539 

5551 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

39 

59" 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

47 

^.r' 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

691 1 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

53 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

"So, 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

LOGARITHMS 


251 


No. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 
9 

No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

252  COLLEGE  ALGEBRA 

199.   Numerical  computation  by  means  of  logarithms.  —  It 

is  of  great  importance  for  the  student  to  be  able  to  perform 
fairly  complicated  numerical  computations  with  accuracy. 
He  must  not  think  that  a  numerical  error  is  of  small  conse- 
quence so  long  as  his  work  has  been  done  on  the  correct  prin- 
ciple. In  all  but  the  simplest  computations  he  will  find  it 
worth  while,  in  the  interests  of  accuracy,  to  give  attention  to 
the  proper  arrangement  of  his  work. 

The  following  general  arrangement  is  suggested : 

Find  the  value  of  5.25x89.46, 
17.35  X  38.61 
log  5.25  =  0.7202  log  17.35  =  1.2393 

log  89.46=  1.9516  log  38.61  =  1.5867 

log  5.25  X  89.46  =  12.6718  -  10         log  17.35  x  38.61  =  2.8260 
log  17.35  X  38.61  =    2.8260 
log    5.25x89.46^    9.8458-10 
^  17.35  X  38.61 


5.25  X  89.46 
17.35  X  38.61 


.7012. 


In  this  work  we  changed  the  form  of  log  5.25  x  89.46  by 
adding  and  subtracting  10  in  order  to  avoid  a  negative  form 
for  the  logarithm  of  the  required  quotient. 

It  is  best  to  lay  out  all  the  work  before  looking  up  any 
logarithms. 

Thus,  in  the  preceding  problem  the  work  was  arranged  in  the  follow- 
ing way  before  any  logarithms  were  filled  in : 

log  5.25  =  0.  log  17.35  =  1. 

log  89.46  =  1.  log  38.61  =  1. 

log  17.35  X  38.61  = 


log  5.25 
log  17.35 

X  89.46  = 
X  38.61  = 

lo£r    ^-25 

X  89.46 

^°^  17.35 
6.26 

X  38.61 
X  89.46 

17.35  X  38.61 

It  is  suggested  that  the  student  find  the  value  of  this  frac- 
tion without  the  use  of  logarithms  and  compare  his  result 
with  that  obtained  aboVe. 


LOGARITHMS  253 

The  following  example  illustrates  the  use  of  an  important 
artifice  in  logarithmic  computation : 

Find  v^.OST  log  .05  =  8.6990  -  10. 

.-.  log  VM  =  2.S997  -  i^. 


Now  it  is  inconvenient  to  have  the  fraction  -^  occurring  here,  and  we 
can  avoid  it  by  writing 

log  .05  =  28.6990  -  30. 

Then  log  v^.'OS  =    9.5663-10. 

The  general  procedure  in  such  cases  is  to  arrange  the  loga- 
rithms in  such  a  way  that  the  final  logarithm  will  appear  in 
the  standard  form. 

The  principal  advantage  in  the  use  of  logarithms  for  per- 
forming the  operations  of  multiplication,  division,  and  raising 
to  an  integral  power  depends  upon  one's  facility  and  accuracy 
in  the  use  of  logarithmic  tables.  But  in  the  extraction  of  a 
root  there  is  a  further  important  and  obvious  advantage.  See 
for  example  the  illustration  on  page  243. 

EXERCISES    AND    PROBLEMS 

Perform  the  following  indicated  operations  by  the  use  of 
logarithms  : 


1.  .372x4.26x37.1.  9.  (6.21)'(9.432)5-- (46.74)1 

2.  98.41x39.72.  3/3.96x4.87 

3.  (9.23)2(.761)3.  *  \13.9x  (5.67)2* 

4.  1246  X  923.4  X. 001672.  ii.  (.8762)2(9.436)2 x (.7582)5. 
5-  ^2.  12^  V6943. 


6.   9762 -^  48.75. 


13.  ^^^2 


3762  X  7931  7963 

8.    ^100.  *   ^4.932 


254  COLLEGE  ALGEBRA 

(12)3(496)'^  X  976.4  17.  ViO. 

*  5621 X  498.6  x  71.34'  ^g    ,^/j4 

3/19.3  x(47.21)^(5.931)^  ^^-  ^2^- 

\  4500  20.  i/iU 


16. 


4500  20.  V(468/. 


21.  What  is  the  weight  in  tons  of  a  marble  sphere  2i  feet 
in  diameter  if  a  cubic  foot  of  water  weighs  62.36  pounds  and 
the  speciJBc  gravity  of  marble  is  2.7  ? 

22.  When  a  weight  of  m  grams  is  attached  to  the  free  end 
of  a  suspended  brass  wire,  the  wire  stretches  S  centimeters. 
If  I  is  the  length  and  r  the  radius  of  the  wire  in  centimeters, 
g  =  980,  and  k  =  1051  .  10^  then 

irr'k 
Compute  S  when  m  =  750,  I  =  164,  and  r  =  .4. 

23.  What  will  $850  amount  to  in  10  years  at  4  %,  interest 
being  compounded  annually  ? 

This  example  affords  a  good  illustration  of  the  limitations  of  a  four- 
place  logarithmic  table. 

We  have  A  -  850  (1.04)io,  and  therefore  log  A  =  log  850+  10  log  1.04. 
Now  all  that  we  can  be  sure  of  in  a  four-place  table  is  that  no  logarithm 
can  be  more  than  .00005  too  much  or  too  little.  Accordingly  in  this  ex- 
pression log  850  is  subject  to  an  error  not  greater  than  .00005  and  10  log 
1.04  to  an  error  not  greater  than  .0005,  and  therefore  log  A  as  computed 
in  this  way  cannot  be  more  than  .00005  -\-  .0005(=  .0006)  too  much  or  too 
little.  The  computed  value  of  log  A  is  .3.0994  and  hence  its  true  value  lies 
between  3.1000  and  3.0988.  The  true  value  of  A  therefore  lies  between 
$  1260  and  1 1255,  while  the  computed  value  is  1 1257.  A  six-place  table 
shows  that  the  true  value  of  A  lies  between  1 1258.21  and  $  1258.18.  A 
seven-place  table  would  give  A  correct  to  the  nearest  cent.  In  practice  the 
nature  of  the  problem  determines  the  number  of  places  that  should  be 
given  in  the  table  used.  The  greater  the  number  of  places  given  in  the 
table  the  greater  the  accuracy  of  the  computations. 

24.  What  sum  will  amount  to  $1500  in  5  years  at  3%,  in- 
terest being  compounded  annually  ? 


LOGARITHMS  255 

25.  If  a  string  weighing  .0098  grams  per  centimeter  is 
stretched  between  two  bridges  60  centimeters  apart  by  a  weight 
of  10  grams,  how  many  vibrations  will  it  make  per  second  ? 

'  (See  Ex.  1,  p.  105.) 

26.  The  weight  w  in  grams  of  a  cubic  meter  of  aqueous  vapor 
saturated  at  15°  is  given  by  the  formula 

^^  ^  1293  X  12.7  X  5 

^  ,  ""      (1  +  J^%)760x8- 

Compute  w. 

27.  The  volume  v  in  litres  of  three  kilograms  of  mercury  at 
85°  is  given  by  the  formula 

13.6  V       5550y 
Compute  V. 

28.  The  time  t  of  oscillation  of  a  pendulum  of  length  I  centi- 
meters is  given  by  the  formula 

Find  the  time  of  oscillation  of  a  pendulum  74.36  centimeters 
long. 

29.  What  must  be  the  length  of  a  pendulum  in  order  that  its 
time  of  oscillation  be  one  second  ? 

30.  The  velocity  -y  of  a  body  that  has  fallen  s  feet  is  given 
by  the  formula  /,..  .. 

What  is  the  velocity  acquired  by  a  body  falling  29  feet 
7  inches  ? 

200.  Exponential  equations.  —  An  exponential  equation  is  an 
equation  that  involves  the  unknown  in  an  exponent. 

The  simplest  exponential  equations  can  be  solved  by 
inspection. 

For  example,  if  4==  =  16, 

it  is  immediately  obvious  that       a  =  2. 


256  COLLEGE  ALGEBRA 

Many  exponential  equations  that  cannot  be  solved  by  inspec- 
tion can  readily  be  solved  by  the  use  of  logarithms  in  the  way 
illustrated  in  the  following  example : 

Solve  2''  =  5. 

Since  the  logarithms  of  equal  numbers  must  be  equal,  we  can  equate 
the  logarithms  of  the  two  members  of  this  equation.     In  this  way  we  get 

log  2=«  =  log  5, 
or  ic  log  2  =  log  5. 

Hence  ^  =  f^g^  ^  ..6990 ^ 2.32. 

log  2      .3010 

Note.  — The  student  should  observe  the  difference  between  -^^  and 
,5  log  2 

log-. 

EXERCISES  AND   PROBLEMS 
Solve  the  following  equations  : 

1.  3^  =  4.  3.   (1.03)^  =  2.  6.   (1.02)^  =  5. 

2.  5^  =  126.  4.   6'='  =  21.  6.    72''+i  =  12. 

7.  In  how  many  years  will  a  sum  of  money  double  itself  at 
3  %,  interest  being  compounded  annually  ? 

In  one  year  $  1  will  amount  to  $  1.03  ;  at  the  end  of  two  years  the  amount 
will  be  1.03  x  1.03  =(1.03)2;  and  at  the  end  of  x  years  the  amount  of 
%  1  will  be  {\my.    Hence  x  must  be  such  that  (1.03)*  =  2.    (See  Ex.  3.) 

8.  In  how  many  years  will  a  sum  of  money  double  itself  at 
3  %,  interest  being  compounded  semiannually  ? 

9.  In  how  many  years  will  $  1000  amount  to  %  1338  SitQfoj 
interest  being  compounded  annually  ? 

If  p  dollars  amount  to  a  dollars  in  n  years  at  r%,  compounded  an- 
nually, then 

«  =  K'  +  i5o)- 

10.  Solve  this  equation  for^^,  r,  and  n  in  turn. 

When  solving  for  n,  and  for  r  when  7i  >  2,  it  is  best  to  use  logarithms. 

11.  What  must  be  the  rate  of  interest  in  order  that  $2500 
may  amount  to  $2814.00  in  four  years,  interest  being  com- 
pounded annually  ? 


CHAPTER   XVIII 

VARIATION 

201.  Constants  and  variables.  —  Many  of  the  letters  that  we 
use  in  mathematics  represent  only  one  number  throughout  a 
given  discussion.  They  are  called  constants.  A  letter,  on  the 
other  hand,  that  represents  different  numbers  in  one  discussion 
is  called  a  variable. 

Thus,  if  t  represents  the  time  that  has  elapsed  since  a  given  moment, 
it  is  a  variable,  since  this  time  is  changing.  Also  in  the  equation 
y  =  x^-\-Qx-\-^  the  letters  x  and  y  are  variables,  since  they  represent  any 
two  numbers  that  are  related  to  each  other  in  the  way  described  by  the 
equation. 

In  some  problems  two  or  more  variables  are  involved,  and 
in  such  cases  there  is  usually  a  connection  between  the  variables. 
The  particular  form  of  this  connection  depends  upon  the  nature 
of  the  problem,  but  there  are  certain  forms  that  occur  much 
more  frequently  than  others.  These  are  described  in  the 
following  paragraphs. 

If  two  variables  are  so  related  that  their  ratio  is  constant, 
either  one  is  said  to  vary  directly  as  the  other. 

Sometimes  the  word  "  directly  "  is  omitted,  and  we  say  that 
one  of  the  variables  varies  as  the  other.  The  two  expressions 
have  the  same  meaning. 

If  a  body  is  moving  at  a  uniform  rate,  the  distance  passed 

over  varies  as  the  time,  since  -  =  a  constant ;  namely,  the  rate. 

The  symbol  oc  when  placed  between  two  numbers  indicates 
that  one  varies  as  the  other.  Thus,  doct  means  that  d  varies 
as  t^  or  that-  =  A;,  where  k  is  a  constant. 

257 


258  COLLEGE  ALGEBRA 

If  two  variables  are  so  related  that  their  product  is  constant, 
either  one  is  said  to  vary  inversely  as  the  other.  If  x  varies 
inversely  as  y,  it  varies  directly  as  the  reciprocal  of  y,  since 

if  xy  —  k,  then^  =  fc,  or  x  =  -.     We  can  accordingly  express 

that  X  varies  inversely  as  y  by  the  symbol  icoc-. 

If  a  body  moving  at  a  uniform  rate  of  r  feet  per  minute 
goes  a  distance  of  d  feet  in  t  minutes,  then  d  =  rt.  In  order 
to  get  over  the  distance  of  d  feet  in  a  different  time  the  rate 
must  be  changed  in  such  a  way  that  the  product  of  the  new 
rate  and  the  new  time  still  equals  d.  Hence,  under  these  cir- 
cumstances, the  rate  varies  inversely  as  the  time. 

If  three  variables  are  so  related  that  the  ratio  of  the  first 
one  to  the  product  of  the  other  two  is  constant,  the  first 
one  is  said  to  vary  jointly  as  the  other  two.     Thus,  if  x  varies 

jointly  as  y  and  z,  then  —  =  k,  ov  x  =  kyz,  where  A;  is  a  constant. 

yz 

The  first  variable  is  said  to  vary  directly  as  the  second  one 
and  inversely  as  the  third  if  it  is  equal  to  the  product  of  a 
constant  and  the  ratio  of  the  other  two.   Thus,  x  varies  directly 

as  y  and  inversely  as  z  if  x  =  -^,  where  Z:  is  a  constant. 

If  the  first  variable  is  equal  to  the  product  of  a  constant  and 
the  ratio  of  the  second  to  the  square  of  the  third,  it  is  said  to 
vary  directly  as  the  second  and  inversely  as  the  square  of  the  third. 

In  this  case  x  =  r^.     Thus,  the  attractive  force  that  draws 

z^  ' 

two  particles  of  matter  together  varies  directly  as  the  product 
of  their  masses  and  inversely  as  the  square  of  the  distance 
between  them. 

It  is  common  in  physics  to  meet  with  two  variables  so  re- 
lated that  one  varies  inversely  as  the  square  of  the  other.     In 

k 
this  case  x  =  -. 


VARIATION  259 

Example.  — 11  xccy,  what  is  the  value  of  x  when  y  =  20? 

We  are  not  given  enough  information  about  the  relation  between  x  andy 
to  enable  us  to  answer  this  question,  for  all  we  are  told  is  that 

x  =  ky, 
where  k  is  some  constant  whose  value  is  not  given. 

But  if  we  are  given  the  additional  information  that  x  =  10  when  y  =  8, 
we  can  find  the  value  of  k,  and  then  answer  the  question. 

Since  x  =  ky, 

and  ic  =  10  when  ?/  =  8,  we  have 

10  =  8  k, 
and  k  =  ^. 

Hence  x  =  ky, 

and  therefore  a;  =  25  when  y  =  20. 

PROBLEMS 

1.  If  X  varies  inversely  as  y  and  x  =  4  when  y  =  9,  what  is 
the  value  of  x  when  y  =  15? 

2.  If  X  varies  jointly  as  y  and  z  and  x=7  when  y  =  5  and 
z  =  12,  what  is  the  value  of  x  when  ?/  =  14  and  z  =  22? 

3.  If  ic  Qc  — ,  and  x  =  15  when  y  =  4  and  2;  =  14,  what  is  the 

z^ 

value  of  x  when  y  =  20  and  2;  =  18  ? 

4.  If  x  varies  inversely  as  the  square  of  y,  what  will  be 
the  effect  on  y  of  doubling  the  value  of  a;  ? 

5.  The  distance  passed  over  by  a  body  falling  from  rest 
varies  directly  as  the  square  of  the  number  of  seconds  during 
which  it  has  fallen,  and  in  3  seconds  it  falls  approximately 
144.9  feet.     How  far  will  it  fall  in  5  seconds  ? 

6.  The  velocity  acquired  by  a  body  falling  from  rest  varies 
directly  as  the  number  of  seconds  during  which  it  has  fallen, 
and  its  velocity  at  the  end  of  2  seconds  is  64.4  feet  per  sec- 
ond.    What  is  its  velocity  at  the  end  of  6  seconds  ? 

7.  The  weight  of  an  object  above  the  surface  of  the  earth 
varies  inversely  as  the  square  of  its  distance  from  the  center 
of  the  earth.  If  an  object  weighs  50  pounds  at  the  sea  level, 
what  would  be  its  weight  on  top  of  a  mountain  a  mile  high  ? 
Assume  that  the  radius  of  the  earth  is  4000  miles. 

I  '5 


± 


260  COLLEGE  ALGEBRA 

-j  8.  When  an  object  is  taken  below  the  surface  of  the  earth 
its  weight  varies  directly  as  its  distance  from  the  center  of 
the  earth.  If  an  object  weighs  200  pounds  at  the  surface, 
how  much  would  it  weigh  midway  between  the  center  and  the 
surface  ?  How  much  would  it  weigh  at  the  center  ? 
^  9.  Would  an  object  that  weighs  100  pounds  at  the  surface 
of  the  earth  weigh  more  5  miles  above  the  surface  than  it  does 
5  miles  below  the  surface,  or  less  ? 
J  10.  The  time  of  vibration  of  a  simple  pendulum  varies  di- 
rectly as  the  square  root  of  its  length.  If  the  time  of  vibra- 
tion of  a  pendulum  39.14  inches  long  is  one  second,  how  long 
must  a  pendulum  be  in  order  that  its  time  of  vibration  shall 
be  two  seconds  ? 

^^  11.  A  pendulum  two  meters  long  makes  61650  vibrations  in 
a  day.  Find  the  length  of  a  pendulum  whose  time  of  vibra- 
tion is  one  second  ? 

12.  Represent  graphically  the  relation  between  two  vari- 
ables when  one  varies  directly  as  the  other. 

13.  Represent  graphically  the  relation  between  two  vari- 
ables when  one  varies  inversely  as  the  other. 

14.  Represent  graphically  the  relation  between  two  vari- 
ables when  one  varies  directly  as  the  square  of  the  other. 

15.  The  amount  of  light  received  on  a  page  from  a  given 
source  varies  directly  as  the  size  of  the  page  and  inversely  as 
the  square  of  its  distance  from  the  source  of  the  light.  One 
page  is  twice  as  large  as  another  one  and  twice  as  far  from  the 
source  of  the  light.     Which  page  receives  the  more  light  ? 

16.  A  metallic  sphere  whose  radius  is  3  inches  weighs  32 
pounds.  How  much  will  a  sphere  of  the  same  material  whose 
radius  is  5  inches  weigh,  if  the  volume  of  a  sphere  varies  di- 
rectly as  the  cube  of  its  radius  ? 


CHAPTER   XIX 

INFINITE   SERIES 

202.   Limit  of  a  variable.  —  Consider  the  sum  of  the  first  n     U 
terms  of  the  geometric  progression  whose  first  term  is  1  and  y        * 
whose  common  ratio  is  ^. 

s  =  Lziffi!  =  2  —  m»-\ 

"  1_JL  ^        \2)       '  /  —  ^ 

The  value  of  s„  depends  upon  the  number  of  terms  repre- 
sented by  it ;  that  is,  upon  the  value  of  n.  Hence  if  we  give 
different  values  to  n,  s„  becomes  a  variable. 

If  we  fix  in  mind  any  small  positive  number,  say  tot^ttto"? 
an  easy  computation  will  show  that  s„  differs  from  2  by  less 
than  this  number  if  n  =  21,  and  the  difference  will  also  be  less 
than  To  oTo  ou"  ^^^  ^^^  greater  values  of  n.  It  is  immaterial  how 
small  the  positive  number  is  that  we  have  in  mind.  As  soon 
as  we  fix  upon  it,  we  can  take  enough  terms  of  the  series  so 
that  the  sum  of  these  terms  or  of  any  greater  number  differs 
from  2  by  less  than  this  number. 

We  express  these  facts  by  saying  that  2  is  the  limit  of  s^  as 
n  increases  without  limit. 

The  relation  between  s„  and  2  can  be  exhibited  graphically 
by  considering  the  successive  values  of  s„  as  abscissae  of  points. 
Of  course  we  cannot  mark  all  of  these  points,  since  there  is  an 
unlimited  number  of  them.  But  a  few  of  the  first  ones  are 
sufficient  to  indicate  that  they  are  getting  successively  nearer 
and  nearer  to  2. 


0  1  li  2 

H h- 1 1— HH 


If 

l2_generalj  we  say  that  a  variable  x  approaches  a  constant  fe 
as  a  limit  if  its  law  of  variation  is  such  that,  when  we  fix  in 

261 


262  COLLEGE  ALGEBRA 

mind  any  positive  number  whatever,  the  difference  between  x 
and  k  will  become  and  remain  less  in  ahsolute  value  than  this 
lmml5er. 

The  student  should  observe  the  importance  of  the  word 
remain  in  this  definition. 

In  the  preceding  illustration,  for  example,  the  absolute  value 
of  the  difference  between  s„  and  Ifi  becomes  less  than  any 
positive  number  we  can  think  of.  As  a  matter  of  fact,  it 
actually  becomes  0  when  n  =  6.  But  when  n  increases  from  G 
the  absolute  value  of  this  difference  increases  and  cannot  for 
any  following  value  of  x  be  less  than  -^^.  We  therefore  do  not 
say  that  s„  approaches  If^  as  a  limit. 

It  follows  from  what  has  just  been  said  that  a  variable  can 
have  only  one  limit. 

The  student  should  note  that  the  limit  of  a  variable  is  al- 
ways a  constant. 

In  the  illustration  the  variable  is  increasing  and  is  therefore 
always  less  than  its  limit.  But  some  variables  are  decreasing 
and  therefore  always  greater  than  their  limits. 

If,  for  example,  S  represents  the  area  of  a  regular  polygon  circum- 
scribed about  a  circle,  and  if  we  increase  without  limit  the  number  of 
sides  of  the  polygon,  /S'  is  a  variable  which  decreases  toward  the  area  of 
the  circle  as  its  limit. 

Still  other  variables  are  increasing  part  of  the  time  and  de- 
creasing part  of  the  time. 

For  example,  the  sum  s„  of  the  first  n  terms  of  the  geometric  progres- 
sion whose  first  term  is  1  and  whose  common  ratio  is  —  ^. 

Here  the  limit  is  |,  and  s„  is  alternately  greater  than  and  less  than 
the  limit. 

Some  variables  do  not  approach  any  limit,  as,  for  example, 
the  variable  t  described  in  §  201. 


INFINITE  SERIES  263 

203.  Convergent  and  divergent  series.  —  A  series  with  an 
unlimited  number  of  terms  is  called  an  infinite  series. 

If  the  sum  of  the  first  n  terms  of  an  infinite  series  approaches 
a  limit  as  n  increases  without  limit,  the  series  is  said  to  be 
convergent. 

The  geometric  progression  referred  to  in  §  202  is  a  conver- 
gent series,  and  in  general  any  geometric  progression  with  an 
unlimited  number  of  terms  and  a  common  ratio  greater  than  —1 
and  less  than  1  is  a  convergent  series  (see  §  100).  But  not  e very- 
convergent  series  is  a  geometric  progression,  as  we  shall  see 
in  §  206. 

If  as  n  increases  without  limit,  s„  does  not  approach  a  limit, 
the  series  is  said  to  be  divergent. 

Example  1.  — Consider  the  series 

1  +  2  +  3  +  --+W  +  ..-. 

If  we  think  of  any  positive  number  M,  no  matter  how  large,  s„  >  M 
for  all  values  of  n  that  are  greater  than  M.  Hence  s„  does  not  approach 
a  limit  and  the  series  is  divergent. 

Example  2.  —  Consider  the  series 

1_1  +  1_1+  ...  ^(_l)n+l+   .... 

Here  Sn  is  alternately  1  and  0  for  successive  values  of  n  and  therefore 
does  not  approach  a  limit.     Hence  the  series  is  divergent. 

There  is  an  important  difference  between  these  two  series.  In  the 
first  one  Sn  can  be  made  as  great  as  we  please  by  taking  n  sufficiently 
great,  while  in  the  second  one  Sn  never  exceeds  1. 

204.  The  general  term  of  a  series.  —  An  infinite  series  is 
not  fully  described  until  we  are  told  how  to  form  any  given 
term.  This  information  is  usually  supplied  by  means  of  the 
nth,  or  general,  term.  It  is  important  therefore  that  the  student 
should  be  able  to  write  down  any  term  of  the  series  when  the 
general  term  is  given. 

Example.  —  Write  down  the  first  five  terms  of  the  series  whose  nth  term 
"2»(2Ll)-     These  terms  are: 

1  ^  1  1       and      1 


2-l'4.3«.58.7  10-9 


264  COLLEGE  ALGEBRA 

EXERCISES 

Write  down  the  first  three  terms  and  the  (ri  +  l)th  term  of 
the  series  whose  nth.  term  is  : 


6. 


1. 

1 
2n 

2. 

1 

3. 

1 

n{7i  +  l) 

4. 

1 

nl 

5. 

n 
3«' 

2  71-1 


—    7.    — 


71 


8    1(^-2)-, 
n       2» 


9.    (-1)» 
10.    (-1) 


/v.2n-l 
+1         ^ 


(2  n  -  1) ! 

n+1  _'fi__ 

(2n)!* 


SERIES  ALL   OF  WHOSE  TERMS  ARE  POSITIVE 

205.  The  only  problem  in  connection  with  infinite  series 
that  will  be  considered  in  this  book  is  that  of  determining 
whether  a  given  series  is  con-vei-gent  or  divergent.  This  is  an 
extremely  difficult  problem,  and  we  shall  discuss  only  the 
simpler  phases  of  it. 

We  shall  suppose  at  first  that  the  series  we  are  dealing  with 
have  all  their  terms  positive,  and  we  shall  make  use  of  the 
following 

Fundamental  principle.  —  If  a  variable  always  i7icreases  (o?-  at 
least  never  decreases)  a7id  never  gets  greater  than  a  given  nu7nher 
M,  then  it  approaches  a  limit  which  is  either  M  or  less  tha7i  M. 

206.  Tests  for  convergence  and  divergence.  —  The  following 
four  tests  for  series  with  positive  terms  are  the  simplest  and 
most  important  ones. 

I.   Comparison  test  for  convergence.  —  Let 

«i  -I-  «2  H +«„+••• 


INFINITE  SERIES  265 

he  an  infinite  series  with  positive  terms.  If  a  second  infinite  series 
with  positive  terms     ,,    ,  .     ,  ,  %.    ,  \ 

is  convergent,  and  if  every  term  of  this  latter  series  is  greater  than, 
or  equal  to,  the  corresponding  term  of  the  first  series,  then  the  first 
series  is  convergent. 

Denote  by  s„  and  S^  the  sums  of  the  first  n  terms  of  these 
two  series  respectively.  As  n  increases  without- limit  S^,  by 
hypothesis,  approaches  a  limit.     Call  this  limit  B. 

Since  the  series  have  positive  terms,  s^  increases  as  n  in- 
creases, but  for  no  value  of  n  is  it  greater  than  S„,  which  in 
turn  is  always  less  than  B.  Hence  s^<B  for  all  values  of  n, 
and  we  know  from  the  fundamental  principle  that  it  approaches 
a  limit.     The  series  is  therefore  convergent. 

Example  1.  —  Consider  the  series 

1  + Jl  +  JL+  ...  +JL  +  ....  (1) 

2!      31  nl  ^^ 

We  know  that  the  series 

1  +  1  +  ^+ -+2^1+-  (2) 

is  convergent.  The  first  two  terms  of  series  (1)  are  the  same  as  the  first 
two  terras  of  series  (2),  but  the  third  term  of  series  (1)  is  less  than  the 
third  term  of  series  (2)  since  K^.  In  general,  the  nth  term  of  series 
(1)  is  less  than  the  nth  term  of  series  (2)  when  n  >  2,  since, 

1      <       1 


1.2.3.  ..w  2.2.  ..2 


(w  —  1)  factors 
Hence  series  (1)  is  convergent. 

Example  2.    Consider  the  series 

1  +  1+J_^_...+ 1 4....  (3) 

After  its  first  term  this. series  is  the  same  as  series  (1).  Hence  if  s„  and 
Sn  denote  the  sums  of  the  first  n  terms  of  series  (1)  and  series  (3)  respec- 
tively, we  have  o       1    .  « 


266  COLLEGE  ALGEBRA 

Now  we  saw  in  Ex.  1  that  as  n  increases  without  limit,  s„,  and  therefore 
also  Sn-i,  approaches  a  limit.  Hence  Sn  approaches  a  limit  and  series  (3) 
is  convergent. 

In  considering  the  convergence  of  a  series  it  is  frequently 
desirable,  as  in  Ex.  2,  to  investigate  the  new  series  obtained 
by  dropping  off  the  first  few  terms  of  the  given  series.  Sup- 
pose that  we  drop  off  the  first  r  terms  and  that  the  sum  of 
these  terms  is  s.  Then  if  S^  and  s^  denote  the  sums  of  the 
first  n  terms  of  the  given  series  and  the  new  series  respectively, 

we  have  S^=s-{-s^_„ 

for  values  oi  n>  r. 

Now  s  is  a  constant,  and  therefore  either  one  of  the  variables 
Sn  and  s„_^  approaches  a  limit  if  the  other  one  does.  Hence 
the  two  series  are  either  both  convergent  or  both  divergent. 

The  stiident  is  reminded  that  s„  and  s„_^,  where  r  is  a  constant, 
represent  different  stages  of  the  same  variable. 

The  conclusion  just  reached  applies  also  to  series  some  of 
whose  terms  are  negative. 

In  order  to  use  Test  I  we  must  know  some  convergent  series 
with  positive  terms  in  order  to  have  a  basis  for  comparison ; 
and  the  more  of  these  we  know  the  wider  the  range  of  usefulness 
of  this  test  will  be. 

EXERCISES 

Show  that  the  following  series  are  convergent ; 

1      . 


1 

1- 

1    1 

1     1 

(2!/ 

(3!/ 

1      ' 

2. 

hh- 

-^. 

+ 

+ 


(niy 


•  2"^l  +  22  "^1-1-33  "^'*"*'l  +  n" 


\\o       \^ 


INFINITE  SERIES  267 

U.    7.   i_4.i_| 1 ± 1 . 

'  3!      5!  {2n  +  l)l 


V 


3      4-2!      5-3!  (n  +  2)-nl 

207.     II.   Comparison  test  for  divergence.  —  Let 

ai  +  «2H !-«„+  ••• 

be  an  infinite  series  with  positive  terms.  If  a  second  infinite  series 
loith  positive  terms  i\    ,   ,     ,  ,   r    A 

is  divergent^  and  if  every  term  of  this  latter  series  is  less  than,  or 
equal  to,  the  corresponding  term  of  the  first  series,  then  the  first 
series  is  divergent. 

Denote  by  s„  and  S^  the  sums  of  the  first  n  terms  of  these 
two  series  respectively.  Since  S^  increases  as  n  increases  and 
does  not  approach  a  limit,  it  must  increase  beyond  all  limit 
(see  fundamental  principle).  But  for  any  value  of  n  it  is 
equal  to,  or  less  than,  s„.  Hence  s^  increases  without  limit  and 
the  first  series  is  divergent. 

In  order  to  have  a  useful  basis  for  comparison  in  the  appli- 
cation of  this  test  we  shall  prove  that  the  series 


is  divergent. 


1  +  -  +  -+...  +-  + 


268  COLLEGE  ALGEBRA 


Consider  the  n  successive  terms  of  this  series, 

1+  1  +...+   1 


n      n  +  1  2?i  —  1 

Every  one  of  these  terms  is  greater  than  — -  and  their  sum 

11  ^^ 

is  therefore  greater  than  n  •  -—  =  -<,     Now  the  series  contains 

2n      2 

an  unlimited  number  of  groups  of  terms  like  this  with  no  terms 
in  common.  If  then,  we  take  n  so  great  that  we  can  form  ten 
million  groups  of  this  kind,  s„  will  exceed  five  million ;  and  by- 
taking  n  sufficiently  large  we  can  make  s„  exceed  any  number 
that  we  had  in  mind.  Hence  s„  does  not  approach  any  limit 
and  the  series  is  divergent. 

This  series  is  known  as  the  harmonic  series  (see  §  101). 

We  are  now  ready  to  illustrate  the  application  of  Test  II. 

Example.  —  Consider  the  series 

l  +  -A_4._A.  +  ...+_2jLzJ_+....  'h 

1.22.3-Tw(«-1)  i 

We  observe  that         JjLnl_  =  ^  n  -  1\  \_ , 
71  {n  —  1)       w  —  1      n 

that  is,  that  the  ?ith  term  of  this  series  is  equal  to times  the  nth. 

71—1 

term  of  the  harmonic  series  for  all  values  of  n  greater  than  1. 

But  when  w>l,  >  1.      Hence,  by  the  comparison  test,  this 

series  is  divergent.       ** "~ 

III.  If  the  nth  term  of  a  series  does  not  approach  zero  as  n  in- 
creases without  limit,  the  series  is  divergent. 

If  a„  is  the  rith  term  of  the  series,  then 

«n  =  Sn  -  S«-l  ; 

and  if  the  series  is  convergent,  s„  and  s„_i  approach  the  same 
limit  as  n  increases  without  limit.  Hence  a„  approaches  the 
limit  zero. 

If  a„  approaches  the  limit  zero  as  n  increases  without  limit, 
it  does  not  follow  that  the  series  is  convergent.  The  harmonic 
series,  for  example,  is  divergent. 


^^ 


INFINITE   SERIES  269 


EXERCISES 
Show  that  the  following  series  are  divergent : 

2.  l  +  2!  +  3!H-. ..  +  %!  +  •••. 

3.  l  +  -L  +  -i4---  +  -i=4--. 
V2      V3  Vw 

4.  2+1  +  1  +  ...  +  '-^  +  .... 
2     4  2w 

fi     ^.L^-l..    ..L^  +  n 
2        O  1  +  71^ 

208.  Since  in  the  application  of  the  next  test  we  shall  have 
occasion  to  look  for  the  limit  of  certain  fractions,  we  shall  ex- 
plain here,  by  means  of  examples,  how  to  proceed  in  such 
cases. 

Example  1.  —  Suppose  that  we  want  to  know  what  happens  to  the  frac- 
tion    ^^  "*"     when  n  increases  without  limit. 
3w+  1 

We  observe  in  the  first  place  that  both  the  numerator  and  the  denomi- 
nator increase  without  limit.  But  this  gives  us  no  information  about  the 
value  of  the  fraction,  since  a  fraction  may  have  any  value  except  0,  and 
have  its  numerator  and  denominator  as  large  as  we  please.  If,  how^ever, 
we  first  change  the  form  of  the  fraction  by  dividing  both  the  numerator 
and  denominator  by  n,  we  shall  be  able  to  get  the  information  we  want. 

2  +  5 
2n  +  5 n^ 

3w  +  l~3      l' 

n 

Now  as  n  increases  without  limit  the  numerator  of  this  last  fraction 

approaches  the  limit  2  and  the  denominator  approaches  the  limit  3,  since 

-  and  -  both  approach  zero.     Hence  the  limit  of  the  fraction  is  |. 
n         n  ^ 


270 


COLLEGE  ALGEBRA 


Example  2. — Consider  the  limit  of  ^— ^t — ^  j"      as  n  increases  without 
limit.  "" 


n^  +  Sn  +  S 
2w-6 


1+5+4 


2      6 


The  numerator  of  this  last  fraction  approaches  the  limit  1  and  the  de- 
nominator approaches  the  limit  zero.  Hence  the  fraction  increases 
without  limit. 

Q  J,  4 

Example  3. —  Consider  the  limit  of  ^ — : — ^ r  as  w  increases  with- 


out limit. 


3w-4 


8  Ji2  _  7  u  _  2 


w2  _  7  M  -  2 

§_1 

n     n^ 

''2      7      2 


The  numerator  of  this  last  fraction  approaches  the  limit  zero  and  the  de- 
nominator approaches  the  limit  8.  Hence  the  fraction  approaches  the 
limit  zero. 

EXERCISES 

Find  the  limit  of  each  of  the  following  fractions  as  n  in- 
creases without  limit. 


1. 


2. 


f 


n-f  1 

n 

n  +  2'        \ 


3"' 

3(n+l)2' 


(n+1)! 


(2n-l)!. 
'    (2n  +  l)! 


7. 


8. 


9. 


n  •  n 


10. 


271-1 
2n  +  l* 

n^  +  n-f-3 
n^  —  71  —  3 

7n-f-4 


n^  +  l 


INFINITE  SERIES  271 

209.    IV.   The  ratio  test.  —  If  the  terms  of  the  Irifinite  series 

«i4-«2H 1-«„4-  ••• 

are  all  positive,  and  if  as  n  increases  without  limit  the  ratio  -^ 
approaches  a  limit  I,  the  series  is  * 

(a)  convergent  when  Z  <  1, 
(6)  divergent  when  Z  >  1. 

Ifl  —  1,  the  series  may  be  either  convergent  or  divergent. 

(a)  1<1. 

Select  some  number  r  that  lies  between  I  and  1  and  is  there- 
fore less  than  1.  Since  -^  approaches  the  limit  I  as  n  increases 
without  limit,  we  can  select  a  positive  interger  m  so  large  that 
-^,  for  all  values  of  n  greater  than  m,  differs  from  I  by  an 

amount  less  in  absolute  value  than  r—l.     (§pe  the  definition 
of  limit,  §  202.)  _^ 

For  these  values  of  n       ^^  <  r.  '  l    i  ' 


a„ 


Hence,      ^  <  r,  or  a^+,  <  ra^+,, 


a 


w+l 


Y-t\^ 


a 


~^<rj  or  am+3<rarn+2<'i^ci^+ij 


a 


m+2 


<r,  or  a^+p+i  <  ra„,+^  <  y^otm+i, 


Therefore  every  term  of  the  series 

except  the  first  one,  is  less  than  the  corresponding  term  of  the 
geometric  progression 


272  .  COLLEGE  ALGEBRA 

which  is  convergent,  since  the  common  ratio  r  is  less  than  1  in 
absolute  value.     The  series 

is  therefore  convergent  and  hence  the  original  series  is  con- 
vergent. 

(&)  Z>1. 

As  in  (a),  we  can  select  a  positive  integer  m  so  large  that 
-^,  for  all  values  of  n  greater  than  m,  differs  from  I  by  an 

amount  less  in  absolute  value  than  l—\.    For  these  values  of  w, 

>1. 
Hence,        ^2^1^  or  a^+2>««.+i, 


a«+i 


a. 


>1,    or    a„,+3>«m+2>am+l, 
''m+2 


Since  these  inequalities  hold  for  all  positive  integral  values 
of  p,  the  series  is  divergent  by  Test  III. 

The  harmonic  series  is  an  example  of  a  divergent  series  for 
which  l  —  \\  and  the  series 

J_  +  ^  +  ...  +  _J_  +  ... 
1.22.3  n(7i  +  l) 

is  an  example  of  a  convergent  series  for  which  l  —  \.     That 
this  series  is  convergent  may  be  seen  by  writing  s„  in  the  form : 

•-{'4)-(i-i)— e-.-i,)--^- 


INFINITE  SERIES  273 


Example.  —  Consider  the  series 

l  +  i  +  2Q)2+...+(w-l)(i)«-i-f  ... 


««+i  _      ng-r 


and  the  limit  of  this  ratio  as  n  in- 


«n  (n-l)(i)-l        3(71-1) 

creases  without  limit  is  |.     Hence  the  series  is  convergent. 

^  What  has  been  said  here  about  series  all  of  whose  terms  are 
positive  applies  with  slight,  obvious  modifications  to  series  all 
of  whose  terms  are  negative. 

In  what  way  is  it  necessary  to  modify  the  fundjamental  priticiple  in 
order  that  it  shall  apply  to  a  variable  that  is  always  decreasing  ? 

EXERCISES 

Test  the  following  series  for  convergence  or  divergence : 

Q2  Qn 

2.    1  +  2  +  3+...+      » 


2!      3!      4!  '  (n  +  l)l 

1        2'       3'  n' 

3  -t  I  I    . . .    I  -I-  . . . 

4  2.3      3.4      4.5  (,,  +  l)(n  +  2) 

5.   14-1+1+. ..  +  ii  +  .... 
2      22     23  2" 


6 


1      2 '  Tt?  —  IV 


7.   1  +  2.  |  +  3-.(|)^  +  ... +»(!)»-'  + 


9        2      1      2'            2' 

1                2""'               1 

■  1  .  3  '  3  .  3'     5  .  3'  ' 

'  (2»-l).3»-'  ' 

.  .,|,|,...,L-,.... 

274  COLLEGE  ALGEBRA 

SERIES   WITH   POSITIVE   AND   NEGATIVE   TERMS 

210.  Alternating  series.  —  A  series  whose  terms  are  alter- 
nately positive  and  negative  is  called  an  alternating  series. 

The  following  test  applies  to  these  series : 

V.  An  alternating  series  is  convergent  if  the  absolute  value  of 
every  term  is  less  than  that  of  the  preceding  term  and  if  the  limit 
of  the  nth  term  is  zero  as  n  increases  without  limit. 

Thus  the  alternating  series 

is  convergent. 

We  shall  omit  the  proof  of  this  test. 

211.  Absolutely  convergent  series.  —  The  symbol  \a\  repre- 
sents the  absolute  value  of  a. 

VI.  Tlie  series      «!  +  ag  +  ag  +  •  •  •  +  a,,  +  .  •  • 
is  convergent  if  the  series 

l«il  +  l«2l  +  k3H — +|<X„|H 

is  convergent. 

We  shall  omit  the  proof  of  this  statement. 

If  the  series  whose  terms  are  the  absolute  values  of  the  cor- 
responding terms  of  a  given  series  is  convergent,  the  given 
series  is  said  to  be  absolutely  convergent. 

The  series  l  _  1.  +  i +  (_  i)«+i  J- +  ... 

is  absolutely  convergent  (see  §  206). 

Every  convergent  series  whose  terms  are  all  of  the  same  sign 
is  absolutely  convergent,  but  not  all  convergent  series  are  abso- 
lutely convergent. 


For  example,  the  series 

1 -Ui -.-+(- i)''^-i^+-.. 

2     3  n 


INFINITE  SERIES  275 

212.   VII.   The  general  ratio  test.  —  If  in  the  series 

«i  +  «2H l-«„H , 

whose  terms  may  he  positive  or  negative,  the  absolute  value  of  the 

ratio  -^±^  approaches  the  limit  I  as  n  increases  without  limit,  the 

an 
series  is  (a)     convergent  when  ^  <  1, 

(b)     divergent  when  Z  >  1. 
Ifl  =  l  the  series  may  be  either  convergent  or  divergent. 

(a)  I  <  1. 

By  Test  IV  the  series  is  absolutely  convergent  and  therefore, 
by  Test  VI,  it  is  convergent. 

(b)  I  >  1. 

In  this  case  a^  cannot  approach  0  as  w  increases  without 
limit,  and  therefore,  by  Test  III,  the  series  is  divergent. 

That  the  series  may  be  either  convergent  or  divergent  when 
Z  =  1  is  shown  by  the  examples  cited  under  Test  IV. 

The  general  ratio  test  includes  Test  IV  as  a  special  case. 
There  are,  of  course,  many  other  tests,  which  cannot  be  given 
here. 

EXERCISES 

Test  the  following  series  for  convergence  or  divergence : 
^-  3     4   2  +  5   3  +^     ^>     ;7T2   »+      • 

Q3  05  Q2r»-1 

3!      5!  ^       ^      (271-1)!^ 

Q2  Q4  q2n 


5.      l-A,  +  _L-...+(-l)n+l_L+.... 

V2      V3  -^n 

93  95  92»-l 

3!      5!  ^^       ^      (2n-l)!^ 


.,-.(>.!)■ 


276  COLLEGE  ALGEBRA 

213.    Power  series.  —  A  series  of  the  form 

where  the  c's  are  constants,  is  called  a  power  series. 

A  power  series  may  converge  for  all  values  of  x,  or  it  may 
diverge  for  all  values  of  x  except  0.  These  are  the  extreme 
cases.  Usually  such  a  series  converges  for  some  values  of  x 
and  diverges  for  the  other  values. 

Test  VII  is  the  most  useful  one  for  determining  the  values 
of  X  for  which  a  power  series  converges. 

Example.     Consider  the  series 

^__^+_^_ ...  +  (_  i)n+i — ^^aii — ^  ... 


1      33.3      3S.5  ^       '      32^-1  (2  w-  1) 


Here 


ttw+l 


32»-i(2n-l) 


a;2(2n-l) 
32(2w  +  l) 


32«+i(2n  +  l)  x2«-i 

The  limit  of  this  fraction  as  n  increases  without  limit  is  — . 

32        ^ 

Hence  the  series  is  convergent  for  those  values  of  x  for  which  —  <  1,  or 
for  which  —  3  <  sc  <  3  ;  and  the  series  is  divergent  for  those  values  of  x 

for  which  —  >  1.    This  inequality  is  satisfied  whenever  a:  <  —  3  or  a;  >3. 
32 

The  test  gives  no  information  as  to  the  convergence  or  divergence  of  the 

series  when  x=  —  ^  or  3.     These  cases  require  a  special  investigation. 

It  is  easy  to  see  directly  that  this  particular  series  is  convergent  for  either 

of  these  values  of  x. 

The  values  of  x  for  which  the  series  is  convergent  form  what 
is  called  the  interval  of  convergence  of  the  series. 

Thus,  the  interval  of  convergence  of  the  series  just  considered 
extends  from  —  3  to  3. 

We  can  indicate  our  conclusions  graphically  by  making  the 
interval  of  convergence  heavier  than  the  rest  of  the  axis. 

-3  0  3 

1 1 \ 

Divergent  Convergent  Divergent 


V 


•INFINITE  SERIES  277 


EXERCISES 

Determine  the  interval  of  convergence  of  each  of  the  following 
series,  and  represent  it  graphically  : 

rvA  /v,5  /v.2n-l 


w 


2!      4!  ^       '      {2n)\ 

^3         /V.5  ^.2n-l 


5.    l  +  x  +  x^^ (-«"'^H . 

7.  0,  +  -  +  ^+...+-+.... 

8.  9  +  10^  +  35^  2«l+5»  +  2^_ 


APPENDIX 

FORMULAE  FROM    SOLID    GEOMETRY 

1.  Section  of  a  pyramid.  —  If  a  plane  be  drawn  parallel  to 
the  base  of  a  pyramid  V-ABC,  cutting  the  pyramid  in  the 
section  DEF,  and  if  JSi  and  S2  represent  the  areas  of  ABC  and 
DEF  respectively,  then  „ 


S2      VK' 


where  VH  and  F/rare  the  distances  of  the  base  and  the  cutting 
plane  respectively  from  the  vertex.     (For  figure,  see  p.  37.) 

2.  Surface  and  volume  of  a  sphere.  —  If  S  represents  the 
surface,  and  Fthe  volume,  of  a  sphere  of  radius  r,  then 

8  =  4:177^, 

and  V=^Trr^. 

3.  Volume  of  a  spherical  segment  of  one  base. — If  a  sphere 
be  divided  into  two  parts  by  a  plane,  either  part  is  called  a 
spherical  segment  of  one  base. 

The  section  made  by  the  plane  is  a  circle  and  is  called  the 
base  of  the  segment. 

If  we  draw  the  radius  of  the  sphere  through  the  center  of 
the  base,  the  length  of  that  part  of  the  radius  which  lies  between 
the  base  and  the  surface  of  the  sphere  is  called  the  altitude 
of  the  segment. 

If  V  represents  the  volume  of  the  segment,  x  its  altitude,  and 
ri  the  radius  of  its  base,  then 

i;  =  1.  TTXri^  -\-  i  TTX^. 
278 


APPENDIX  279 


FORMULiE   FROM   PHYSICS 

1.  Distance  passed  over  by  a  falling  body.  —  The  distaDce  s 
passed  over  by  a  falling  body  in  t  seconds  is  given  in  feet  by 
the  formula  —  16  /^ 

2.  Velocity  of  a  falling  body.  —  The  velocity  v  of  a  body  that 
has  fallen  s  feet  is  given  in  feet  per  second  by  the  formula 


i;  =  V64.3  5. 

The  formulae  in  1  and  2  are  both  approximate,  but  the  latter  Is  the 
more  exact. 

3.   Velocity  of  a  projectile.  —  The  velocity  v  of  a  projectile 
at  any  moment  in  feet  per  second  is  given  by  the  formula 


V  =  Vi^o^  —  64  y, 

where  -^o  is  the  initial  velocity  and  y  is  the  height  of  the  pro- 
jectile at  this  moment  in  feet  above  the  level  of  the  starting 
point. 

4.  Number  of  vibrations  made  by  a  stretched  wire.  —  The 

number  n  of  vibrations  made  by  a  stretched  wire  is  given  by 

the  formula  __^^ 

1     /980  M 

where  I  is  the  length  in  centimeters  between  the  bridges,  M  the 
weight  in  grams  of  the  stretching  weight,  and  7n  the  weight  in 
grams  of  the  wire  per  centimeter  of  length. 

5.  Oscillation  of  a  pendulum. — The  time  t  of  oscillation  of 
a  pendulum  of  length  I  centimeters  is  given  in  seconds  by  the 
formula  . 

6.  The  lever.  —  The  point  of  support  of  a  lever  is  called  the 
fulcrum. 


280  APPENDIX 

There  are  different  kinds  of  levers,  but  in  all  of  them  two 
applied  forces  are  in  equilibrium  (the  effect  of  the  weight  of  the 
lever  being  neglected)  when  the  product  of  the  first  force  and 
the  distance  of  its  point  of  application  from  the  fulcrum  is 
equal  to  the  product  of  the  second  force  and  the  distance  of  its 
point  of  application  from  the  fulcrum. 

fA=fA' 

The  two  forces  must  be  applied  in  the  same  direction  when 
their  points  of  application  are  on  opposite  sides  of  the  fulcrum, 
as  in  a  teeter  board.  The  forces  must  be  applied  in  opposite 
directions  when  their  points  of  application  are  on  the  same 
side  of  the  fulcrum. 

^'        —d^      — \  y^ 

When  two  forces  /i  and  f^  act  in  the  same  direction  on  a 
lever  at  the  distances  d^  and  cZg  respectively  from  the  fulcrum 
and  on  the  same  side  of  it,  the  combined  effect  is/iC?i  +/2C?2. 

The  effect  of  the  weight  of  a  uniform  lever  is  the  same  as  if 
the  entire  weight  were  concentrated  at  the  middle  point. 

7.  Center  of  gravity. — If  the  centers  of  gravity  of  two  bodies 
of  weights  TFi  and  W^  respectively  lie  on  the  ic-axis  and  have  the 
abscissae  a  and  h  respectively,  then  the  abscissa  c  of  the  center 
of  gravity  of  the  two  together  is  given  by  the  formula 

8.  Specific  gravity. — The  ratio  of  the  weight  of  a  volume  of 
a  given  substance  to  the  weight  of  the  same  volume  of  water 
is  called  the  specific  gravity  of  the  substance. 

Thus,  a  cubic  inch  of  platinum  weighs  approximately  22  times  as  much 
as  a  cubic  inch  of  water,  and  we  accordingly  say  that  the  specific  gravity 
of  platinum  is  approximately  22. 

9.  Principle  of  Archimedes.  —  A  body  immersed  in  a  liquid 
loses  a  part  of  its  weight  equal  to  the  weight  of  the  displaced 
liquid. 


INDEX 

(Numbers  refer  to  pages.) 


Abscissa,  2,  48, 

Absolute  value,  2,  162,  274. 

Addition,  definition  of,  3. 

definition  of,  for  imaginary  num- 
bers, 154,  156. 

Algebraic  solution  of  linear  equations 
in  two  or  more  unknowns,  51. 

Amplitude  of  a  number,  162. 

Antecedent  of  a  ratio,  33. 

Antilogarithm,  239. 

Argument  of  a  number,  162. 

Arithmetic  means,  124. 

Arithmetic  progression,  123. 

Associative  law  of  addition,  3. 

Associative  law  of  multiplication,  7. 

Assumptions  for  addition,  3. 

Assumptions  for  multiplication,  7. 

Axes,  coordinate,  48. 

Axis,  1. 


Base, 

change  of,  242. 

of  system  of  logarithms,  239. 
Binomial  theorem,  141,  150. 


Cancellation,  law  of,  for  addition,  4. 

for  multiplication,  7. 
Circle,  equation  of,  66. 
Coefficients,  relations  between  roots 

and,  176. 
Combinations,  138. 

number  of,  139. 
Complex  numbers,  153,  158. 
Conjugate  numbers,  159. 
Consequent  of  ratio,  33. 
Constants,  definition  of,  257. 
Continuation  of  signs,  186. 
Convergent  series,  definition  of,  263. 
Coordinates  of  a  point,  48. 


De  Moivr6's  Theorem,  163. 
Dependent  equations,  53,  59. 
Descartes's  Rule  of  Signs,  187. 
Determinants,  definition  of,  204. 

elements  of,  204. 

minors  of,  208. 

of  order  n,  201. 

of  second  order,  55. 

of  third  order,  62. 

properties  of,  205-208. 

solution  of  equations  by  means  of, 
216. 

terms  of,  204. 
Difference,  definition  of,  4. 
Discriminant  of  a  quadratic,  91. 
Distributive    law    of    multiplication, 

7. 
Divergent  series,  definition  of,  263. 
Dividend,  definition  of,  8,  16. 
Division,  definition  of,  8,  16. 
Divisor,  definition  of,  8,  16. 

Equations,  definition  of,  38. 

dependent,  53,  59. 

exponential,  255. 

homogeneous,  219. 

inconsistent,  53,  59,  219. 

independent,  60. 

in  form  of  quadratics,  103. 

involving  fractions,  96. 

involving  radicals,  99. 

linear,  38. 

of  condition,  38. 

of  first  degree,  38. 

quadratic,  84. 

rational  integral,  38. 

systems  of,  47,  110. 

transformations  of,  178-183. 

with  given  roots,  85. 
Ellipse,  112. 


281 


282 


INDEX 


Exponents,  fractional,  73. 

fundamental  laws  of,  73. 

irrational,  238. 

negative,  73. 

zero,  75. 
Extremes  of  a  proportion,  33. 

Factorial  n,  136. 
Factor,  rationalizing,  82. 
Factor  Theorem,  170. 
Factors,  7,  19. 
Fourth  proportional,  34. 
Fractions,  definitions  and  principles, 
27. 

partial,  228. 
Function,  rational,  81. 

rational  integral,  81. 
Fundamental    theorem    of    algebra, 
174. 

General  term  in  binomial  expansion, 
143,  152. 

Geometric  means,  128. 

Geometric  progression,  127. 

Graphical  interpretation  of  Trans- 
formation III,  183. 

Graphical  representation,  47. 

Graphical  solution  of  equations,  51, 
93,  95,  110-120,  172. 

Graphical  solution  of  linear  and  quad- 
ratic inequalities,  225. 

Greater  than,  meaning  of,  2. 

Greatest  coeflEicient  in  binomial  ex- 
pansion, 143. 

Harmonic  progression,  133. 
Highest  common  factor,  21. 

Euclid's  Method,  24.  ' 
HoTl^er's  Method,  192. 
Hyperbola,  113. 

Identity,  17,  38. 
Imaginary  numbers,  153,  158. 
Imaginary  roots,  177. 
Inequalities,  definition  of,  221. 

conditional  and  unconditional,  221, 
223. 

properties  of,  222. 
Infinite  geometric  progression,  131. 
Inversions,  of  letters,  202. 

of  numbers,  201. 


Irrational  roots,  Homer's  Method  for 
finding  approximate  values  of, 
192. 

Less  than,  meaning  of,  2. 
Limit,  definition  of,  131,  261. 
Locus  of  an  equation,  50. 
Logarithms,  change  of  base,  242. 

characteristic  of,  243. 

common,  241,  242. 

definition  of,  239. 

mantissa  of,  243. 

natural,  241. 

table  of,  250,  251. 

use  of  table  of,  243. 
Lowest  common  denominator,  30. 
Lowest  common  multiple,  22. 

Mean  proportional,  34. 
Means  of  a  proportion,  33. 
Minor  of  determinant,  208. 
Minuend,  definition  of,  4. 
Modulus  of  a  number,  162, 
Multiplication,  associative  law  of,  7. 

definition  of,  6. 

definition  of,  for  imaginary  num- 
bers, 157,  159. 

Numbers,  commensurable,  33. 
complex,  153,  158. 
equal,  155. 
imaginary,  153,  158. 
incommensurable,  33. 
negative,  2. 
positive,  2. 
pure  imaging^ry,  158. 
rational,  2. 
real,  2. 

representation  of  points  by,  1,  47, 
153. 

Ordinate,  48. 
Origin,  1,  48. 

Parabola,  94,  113. 
Parameter,  116. 
Parentheses,  5. 

removal  and  insertion  of,  10. 
Permutations,  definition  of,  134. 

of  n  things  not  all  different,  137. 

of  n  things  r  at  a  time,  135. 


INDEX 


283 


Polar     representation     of      complex 

numbers,  162. 
Polynomials,   addition   and   subtrac- 
tion of,  12. 

division  of,  16. 

homogeneous,  14. 

multiplication  of,  14. 

prime,  21. 
Principal  root,  74,  160. 
Products,  special,  16. 
Proportion,  by  alternation,  34. 

by  composition,  35. 

by  composition  and  division,  35. 

by  division,  35. 

by  inversion,  34. 

definition  of,  33. 

Quotient,  definition  of,  8,  16. 

Radicals,    addition    and   subtraction 
of,  79. 

coefficient  of,  78. 

index  of,  78. 

multiplication  of,  80. 

radicand  of,  78. 

similar,  79. 

simplification  of,  78,  79. 
Ratio,  definition  of,  33. 

of  geometric  progression,  127. 
Rationalizing  factor,  82, 
Rational  roots  of  an  equation,  168, 

190. 
Ratio  test  for    infinite    series,    271, 

275. 
Real  number,  2,  159. 
Reciprocal  of  a  number,  7. 
Remainder,  definition  of,  in  division, 
16,  17. 

definition  of,  in  subtraction,  4. 
Remainder  Theorem,  169. 
Repeating  decimals,  132. 
Representation  of  points  by  numbers, 

1,  48,  153,  154.    . 
Roots,  of  an  equation,  39. 


Roots,  imaginary,  177. 

irrational,  192. 

negative  irrational,  197. 

number  of,  175. 

positive  irrational,  192. 

rational,  190. 

relation  between,  and  coefficients, 
176. 
Roots  of  numbers,  164,  165. 

principal,  74,  160. 

Series,  absolutely  convergent,  274. 

alternating,  274. 

comparison  test  for  convergence  of, 
264. 

comparison  test  for  divergence  of, 
267. 

convergent,  263. 

divergent,  263. 

general  ratio  test  for,  275. 

general  term  of,  263. 

limit  of  certain  geometric,  131. 

power,  276. 

ratio  test  for,  271. 

with  positive  terms,  264. 

with  positive  and  negative  terms, 
274. 
Sign,  continuations  and  variations  in, 

186. 
Signs,  Descartes's  Rule  of,  187. 
Straight  lines,  equation  of  two,  113, 

114. 
Subtraction,  definition  of,  4. 
Subtrahend,  definition  of,  4. 
Synthetic  division,  170-172. 
Systems  of  equations,  47,  110,  216. 

Tables  of  logarithms,  250,  251. 
Tabular  difference,  249. 
Third  proportional,  34. 

Variable,  definition  of,  257. 

limit  of,  131,  261. 
Variations  in  sign,  186. 


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